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# SEM CLAS POL THINK POL S 240

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NOV 6 2008 LEC 12 ECON 240Al L Phillips Bivariate Normal Distribution Isodensity Curves I Introduction Economists rely heavily on regression to investigate the relationship between a dependent variable y and one or more independent variables x w etc As we have seen graphical analysis often provides insight into these bivariate relationships and can reveal nonlinear dependence outliers and other features that may complicate the analysis There are other methodologies for examining bivariate relations We have examined some of them For example correlation analysis using the correlation coefficient p is one method as discussed in Lecture Eight Another method is contingency table analysis We will discuss the latter shortly First we turn to the bivariate normal distribution which provides a useful visual model for bivariate relationships just as the univariate normal distribution provides a useful probability model for a single variable It is useful to have a mental model in mind for bivariate relationships and the iso density lines or contour lines of the bivariate normal provide a visual representation The bivariate normal distribution of two variables y and x is a joint density function fxy and if the variables are jointly normal then the marginal densities eg fx and fy are each normal In addition the conditional densities y given x fyx are normal as well The isodensity lines ie the locus where fxy is constant is a circle around the origin for the bivariate normal if both x and y have mean zero and variance one ie are standardized normal variates and are not correlated If x and y have nonzero means ux and Hy respectively then these contour lines are circles around the point ux Hy If x has a larger variance than y then the contour lines are ellipses with the long axis in the x direction If x and y are correlated then these ellipses are slanted NOV 6 2008 LEC 12 ECON 240A2 Bivariate Normal Distribution Isodensity Curves L Phillips 11 Bivariate Normal Density The density function fXy for two jointly normal variables X and y where for z example X has mean ux variance 6x and correlation coef cient p is x y 12ncx 6y V0125 eXp121pzx Hx6x2 2px Hx6x y way y My6yz 1 A Case 1 correlation is zero means are zero and variances are one x y 12n exp12 x2 yz 2 and for an isodensity where fXy is a constant k taking logarithms In 211 fx y 12 x2 yz or x2 yz 2 In 211 fX y 21n 211 k 3 Recall X2 yz r2 is the equation of a circle around the origin 0 0 with radius r as illustrated in Figure 1 Figure 1 Isodensity Circles About the Origin NOV 6 2008 LEC 12 ECON 240A3 L Phillips Bivariate Normal Distribution Isodensity Curves Note that if X and y are independent then the correlation coefficient p is zero and the joint density function fX y is the product of the marginal density functions for X and y ie fX y fX fy 145 eXp 12 x2 15 eXp 12 yz 4 where X and y have mean zero and variance one B Case 2 correlation is zero variances are one means ux and Hy In this case the origin is translated to the point of the means ux Hy The bivariate density function is x y 12n exp 12x ux2 y M 1 5 For a density equal to k x m2 y My 1 2 In 21 fxy 2 1n2nk 6 This is illustrated in Figure 2 Figure 2 Isodensity Lines About the Point of Means Bivariate Normal NOV 6 2008 LEC 12 ECON 240A4 L Phillips Bivariate Normal Distribution Isodensity Curves C Case 3 correlation is zero variance ofX gt variance ofy If the variance of X exceeds the variance of y then the isodensity lines are ellipses about the point of the means with the semimajor aXis in the X direction fxy 101 6x 6y exp 12xuxcxz xuyycyf 7 Note that if X and y are independent then the correlation coefficient is zero and the joint density is the product of the marginal densities x y fX fy Max M exp12x Hx6x216y J5 exp12y My6y2 For a constant isodensity fX y k from Eq 7 we have XHx6xz XHy6yz 2 In 211 6x 6y fX y 2 In 211 6x 6y k 8 Recall the equation of an ellipse about the origin with semimaj or aXis a and semiminor aXis b is Xza2 yzbz l 9 Elliptical isodensity lines around the point of the means are illustrated for Eq 7 in Figure 3 Case 4 correlation is nonzero The joint density function is given by Eq 1 above and the isodensity lines are tilted ellipses around the point of the means as illustrated in Figure 4 for positive autocorrelation NOV 6 2008 LEC 12 ECON 240A5 L Phillips Bivariate Normal Distribution Isodensity Curves Hy Figure 3 Isodensity Lines About the Point of the Means Var X gt Var y Hy Figure 4 isodensity lines X and y correlated III Marginal Density Functions If X and y are jointly normal then both X and y each have normal density functions For eXample the marginal density of X fX is NOV 6 2008 LEC 12 ECON 240A6 L Phillips Bivariate Normal Distribution Isodensity Curves fX bawdy 1cx J5 exp12x pro6x12 10 and similarly for y IV Conditional Density Function The density of y conditional on a particular value of X X X is just a vertical slice of the isodensity curve plot at that value of X and if X and y are jointly normal is also normal It can be obtained by dividing the joint density function by the marginal density and simplifying fyx x yfx 1cy J5 1 pal2 exp121pz6yzyHypXHx6y6x 11 where the mean of the conditional distribution is My pXHx6y6x ie this is the eXpected value of y for a given value of X such as X EyXX Hy W uxxcycx 12 So ifX is at its mean X then the eXpected value ofy is its mean Hy IfX is above its mean and the correlation is positive then the eXpected value of y conditional on X is greater than Hy This is called the regression ofy on X with intercept Hy p Hx6y6x and slope p6y6x Of course ifX and y are not correlated then the slope is zero and the intercept is Hy The variance of the conditional distribution is VaryXX 61 pz 13 The isodensity lines and the regression line the mean of y conditional on X is illustrated in Figure 5 for the case where X and y are positively correlated and the variance of X is greater than the variance of y NOV 6 2008 LEC 12 ECON 240A7 L Phillips Bivariate Normal Distribution Isodensity Curves Hy Expected Value of y Conditional Figure 5 The Expected value of y Conditional on x V Example Rates of Return for a Stock and the Market In Lab Six we look at the data le XR1734 for 48 monthly rates of return to the General Electric GE stock and the Standard and Poor s Composite Index Both of these variables are not signi cantly different from normal in their marginal distributions An example is the histogram and statistics for the rate of return for GE shown in Figure 6 Figure6 6 SeriesGE Sample 199301 199612 5 7 39 3 Observations 48 4 Mean 0 022218 Median 0 019524 3 Maximum 0117833 Minimum 0 058824 Sid Dev 0 043669 2 Skewness 0 064629 Kurtosis 2 231861 139 JarqueBera 1213490 6 Hobability 0545122 0 0 b5 060 065 NOV 6 2008 LEC 12 ECON 240A8 L Phillips Bivariate Normal Distribution Isodensity Curves The coefficient of skewness S is a measure of nonsymmetry S1IIZJf Tillt93 14 j1 Where 639 is s the sample standard deviation For the normal distribution the coefficient of skewness is zero since the cube of deviations from the mean sum to zero with the negative values offset by the positive ones because of symmetry The coefficient of kurtosis K is a measure of how peaked or how at the density is capturing the weight in the tails K1n Zyj J7lamp4 15 11 For the normal distribution the coefficient of kurtosis is three The JarqueBera statistic JB combines these two coefficients JB n M6 S2 14K 7 32 16 Where k is the number of estimated parameters such as the sample mean and sample standard deviation needed to calculate the statistics If S is zero and K is 3 then the JB statistic will be zero Large values of JB indicate a deviation from normality and can be tested using the ChiSquare distribution with two degrees of freedom The descriptive statistics for GE and the Index are given in Table 1 The estimated correlation coefficient is 0636 These estimates can be used to implement Eq 12 EyXX Hy p Hx5y5xl pX6y6x EGEIndex 00222 7 0636001440043700254 06361720Index EGEIndex 00064 1094IndeX 13 NOV 6 2008 LEC 12 ECON 240A9 L Phillips Bivariate Normal Distribution Isodensity Curves For comparison the estimated regression is reported in Table 2 The coefficients are nearly identical So the regression can be interpreted as the expected value of y for a given value of X A plot of the rates of return for GE and the stock Index are shown in Figure 6 Table 1 Sample 199301 199612 GE INDEX Mean 0022218 0014361 Median 0019524 0017553 Maximum 0117833 0076412 Minimum 0058824 0044581 Std Dev 0043669 0025430 Skewness 0064629 0453474 Kurtosis 2231861 3222043 JarqueBera 1213490 1743715 Probability 0545122 0418174 Observations 48 48 Table 2 Dependent variable GE Method Least Squares Coefficient Std Error tStatistic Prob 0006526 0005659 1153229 02548 1092674 0195328 5594046 00000 0404865 Mean dependent var 0022218 0391927 SD dependent var 0043669 0034053 Akaike info criterion 3881039 0053341 Schwarz criterion 3803072 9514493 Fstatistic 3129335 2442439 ProbFstatistic 0000001 NOV 6 2008 LEC 12 ECON 240A10 L Phillips Bivariate Normal Distribution Isodensity Curves Fig ure 6 Rates of Return for GE Stockand SampP Composite Index 015 010 7 005 7 GE 000 7 005 7 010 INDEX V Discriminating Between Two Populations As an example we will use the data le XR1858 on lottery expenditure as a percent of income introduced in Lab Six Twentythree individuals did not gamble The means for their age number of children years of education and income are shown in Table 3 For comparison the means of the 77 individuals who did gamble are shown in Table 4 The question is can these explanatory variables predict who will and who will not buy lottery tickets The means for number of children and age are fairly similar for the two groups Those who do not buy lottery tickets are better educated with higher incomes than those NOV 6 2008 LEC 12 ECON 240All L Phillips Bivariate Normal Distribution Isodensity Curves who participate in the lottery The correlation between education and income is 065 for ticket buyers and 074 for the entire sample Table 3 Sample 1 23 AGE CHILDREN EDUCATION INCOME LOTTERY Mean 4043478 1782609 1556522 4756522 0000000 Median 4100000 2000000 1600000 4200000 0000000 Maximum 5400000 4000000 2000000 9500000 0000000 Minimum 2300000 0000000 7000000 1800000 0000000 Std Dev 8805092 1277658 3368653 2251631 0000000 Skewness 0446250 0014659 0919721 0518080 NA Kurtosis 2308389 1985475 3156800 2097295 NA JarqueBera 1221762 0987199 3266130 1809815 NA Probability 0542872 0610425 0195330 0404579 NA Observations 23 23 23 23 23 Table 4 Sample 24 100 AGE CHILDREN EDUCATION INCOME LOTTERY Mean 4419481 1779221 1194805 2854545 7000000 Median 4300000 2000000 1100000 2700000 7000000 Maximum 8200000 6000000 1700000 6400000 1300000 Minimum 2100000 0000000 7000000 1100000 1000000 Std Dev 1270727 1343830 2887797 9423578 2695025 Skewness 0466514 0506085 0293006 1304264 0308533 Kurtosis 3189937 3149919 1918891 5036654 2741336 JarqueBera 2908734 3359008 4851659 3513888 1436299 Probability 0233548 0186466 0088405 0000000 0487654 Observations 77 77 77 77 77 The conceptual framework is provided in Figure 7 which shows isodensity curves for the two populations for the explanatory variables income and education NOV 6 2008 LEC 12 ECON 240A12 L Phillips Bivariate Normal Distribution Isodensity Curves Lottery Players L ottery Av01ders Y education x Hy L x Decision Rule Hy K L jne x in X income Figure 7 Discriminating Between Those Who Play the Lottery and Those Who Don t Using a single variable we could test for a difference in sample means for education or for a difference in the sample means for income But why not use both variables and instead of a decision rule classifying them as gamblers if X lt X or y lt y use a decision rule line that separates the two populations This is called discriminant function analysis Another approach is to use a probability model A linear probability model can be estimated with regression using a dependent variable coded one for those who buy tickets and zero for those who do notdesignated bern for Bernoulli and regressing it against education and income The results are shown in Table 7 with a plot of actual tted and residuals following Since income is very skewed it is better to use the natural logarithm of income which is more bell shaped NOV 6 2008 LEC 12 ECON 240Al3 L Phillips Bivariate Normal Distribution Isodensity Curves Using the same coding for the dependent variable nonlinear estimation of the logit probability model is possible using Eviews which avoids some problems that occur with the linear probability model Table 7 Dependent Variable BERN Method Least Squares Sample 1 100 Included observations 100 Variable Coefficient Std Error tStatistic Prob EDUCATION 0021597 0016017 1348392 01807 INCOME 0010462 0003430 3049569 00030 C 1390402 0148465 9365178 00000 Rsquared 0277095 Mean dependent var 0770000 Adjusted Rsquared 0262190 SD dependent var 0422953 SE of regression 0363299 Akaike info criterion 0842358 Sum squared resid 1280264 Schwarz criterion 0920513 Log likelihood 3911792 Fstatistic 1859045 DurbinWatson stat 0651758 ProbFstatistic 0000000 Figure 8 Actual Fitted and residuals from Linear Probability Mod 710 i Filmm 1M rr4r r H A i l hli li mlill AIMi l l ll iilllMllrjV AWN U lllmp l l H v WH V H y W llwl l I W l l 705 10J l vlw l 1 ll 05 77777 l WOD on 205 05 10 l l l l l l l l l l l l l l l l l l l 10 20 30 40 50 60 70 80 90 10 7 Residual 77777 Actual WW Fitted NOV 6 2008 LEC 12 ECON 240Al4 L Phillips Bivariate Normal Distribution Isodensity Curves The linear probability model can be interpreted from the perspective of decision theory and used to come up with a decision rule or discriminant function The expected cost of misclassification is the sum of the expected costs of two kinds of misclassification l labeling a nonplayer a player and 2 labeling a player a non player For example if we have the cost of labeling a nonplayer a player CPN and multiply it by the conditional probability PPN of incorrectly classifying this non player a player given this individual s values for income and education and multiply by the probability of observing nonplayers in the population PN we have this first component of misclassification CPNPPNPN Adding the other expected cost of misclassification we have the total expected costs EC of misclassification EC CPNPPNPN CNPPNPPP 14 If the two costs of misclassification are equal ie CPN CNP noting that there are 23 nonplayers or about one in four in the population the expected costs are EC CPNPPNl4 CNPPNP34 15 We could weight the expected costs of misclassification equally by setting the probability of classifying a nonplayer coded one in the linear probability model as a player to 3 ie setting 15 PN 3 ie EC CPN34l4 CNPl434 16 This is equivalent to setting the fitted value of Bern to 3A and classifying an individual as a player if the individuals fitted probability is greater than 3 ie if 1 em gt 3A where Bern 3A 1390 700216education 7 00105income l7 drawing on Table 7 Thus the discriminant function or decision rule line in education income space is rearranging Eq 17 4232009 Lecture Eight 1 l Forecasting A For an AR1 process xt b1xt1 et 1 a one period ahead forecast xft as of time t1 is xft Et1xt b1xt1 a The error in this forecast is the difference between the observed value and the forecast forecast error observed value forecast value forecast error xt xft forecast error b1Xt1 et b1Xt1 forecast error et b The variance ofthe forecast error is variance ofthe forecast error var et 62 2 A two period ahead forecast as oftime t1 is xft1 Et1xt1 Note that xt1 b1xt et1 b1b1xt1 et et1 and Et1xt1 b12xt1 a Once again the error in this forecast is the difference between the observed value and the forecast forecast error observed value forecast value 4232009 Lecture Eight 2 forecast error Xt1 Xft1 forecast error b12xt1 b1et et1 b12xt1 forecast error b1et et1 b The variance ofthe forecast error is variance ofthe forecast error var b1et et1 variance ofthe forecast error b12 62 62 1 b12 62 B For an MA1 process Xt a1et1 et 1 a one period ahead forecast Xft as of time t1 is Xft Et1xt a1et1 a The error in this forecast is the difference between the observed value and the forecast forecast error observed value forecast value forecast error Xt Xft forecast error a1et1 et a1et1 forecast error et b The variance ofthe forecast error is variance ofthe forecast error var et 62 2 A two period ahead forecast as oftime t1 is Xft1 Et1xt1 Note that Xt1 a1et et1 4232009 Lecture Eight 3 and Et1xt1 o a Once again the error in this forecast is the difference between the observed value and the forecast forecast error observed value forecast value forecast error Xt1 Xft1 forecast error a1et et1 O forecast error a1et et1 b The variance ofthe forecast error is variance ofthe forecast error var a1et et1 variance ofthe forecast error a12 62 62 1 a12 62 C Calculating Errors for an MA1 Xt a1et1 et or et Xt a1et1 and letting eO equal zero as an initial condition e1 X1 e2 X2 3161 33 X3 3162 etc 4232009 Lecture Eight 4 II The Partial Autocorrelation Function for an MA1 A moving average process of the rst order Xt 30 a1et1 1a1Zet can be inverted and expressed as an infinite autoregressive process 1 a121xt et or 1 a1z a1222 a13Z3 a14Z4 Xt et Xt a1Xt1 a12xt2 a13Xt3 a14xt4 et Recall that the autocorrelation function at lag one ACF1 for an MA1 was a11 a1 which will be the value of the partial autocorrelation function at lag one PACF1 From the formula expressed as an in nite autoregressive process abovethe partial autocorrelation at lag two PACF2 will be a121 a122 the partial autocorrelation at lag three PACF3 will be a131 a123 etc lfthe coef cient a1 is negative then the partial autocorrelation function will have negative values that decay towards zero with lag lll Moving Average Processes of Order Two MA2 A moving average process of order two has the following specification Xt et a1et1 a2et2 with mean funxtion mt equal zero and autocovariance at lag zero vxx0 EXtXt Eet 31601 a2et2et 31601 32602 mO 62 a1262 a22cs2 1 a12 a22 6239 The autocovariance at lag one is mm EXtXt1 Eet 31601 azet2et1 31602 azet3 vXYX1 a162a1a262 a1a1a2 62 4232009 Lecture Eight 5 The autocovariance at lag two is Yxx2 EXtXt2 Eet 31601 azet2et2 31603 32604 Yxx2 a2 62 The autocovariances at higher lags are zero The values of the autocorrelation function at lags one and two are Pxx1 31a1a211 312 322 Pxx2 a21439 312 a22 and pxyxu O u23 Oct 31 2008 LEC 11 ECON 240A1 L Phillips Weibull Distribution Transformations Poisson Distribution 1 Introduction In Lecture Ten we introduced the exponential distribution as a parametric approach to estimating the distribution of time until failure This distribution has one parameter lambda and the reciprocal of lambda is the mean time until failure So the r quot 39 is 1 39 39 in J to estimate but this simplicity came at a price of two assumptions First the hazard rate is constant for the exponential which is restrictive Second the exponential has the no memory feature which means that the survival time to date does not affect the expected time remaining before failure The Weibull is a distribution that permits a little more exibility but at a price of two parameters The survivor function is also nonlinear in these parameters which raises a question about whether we can linearize the function through transformation We can not That leaves the question of how to estimate the equation Lastly we turn to another distribution the Poisson which can be used as an approximation to the binomial for rare events It is useful for modeling problems such as the number of defects on a foot of magnetic recording tape and other applications to quality control 11 Failure Time Models and the Weibull Distribution The Weibull Distribution has the cumulative distribution function Fa 1 7 exptoc 1 And so the survivor function is st17Ft exptoc 2 Taking the derivative of the cumulative distribution function yields the density function dFtdt ft oc mot1 exptoc 3 Oct 31 2008 LEC 11 ECON 240A2 L Phillips Weibull Distribution Transformations Poisson Distribution and so the hazard rate is ha ftst oc mot1 4 Thus the hazard rate is a power function of the duration t If beta equals one then the hazard rate is a constant loc If beta is greater than one then the hazard rate increases with survival time t If beta is less than one then the hazard rate is a decreasing function oft So depending on the value of this parameter three different patterns of behavior for the hazard rate can be explained This does not cover all possibilities such as a situation where the hazard rate may rst increase and then decrease with survival time but it is more exible than the exponential III Transformations We saw in the previous lecture that taking the logarithms of both sides of the equation for the Weibull s survivor function did not result in an equation linear in the parameters We could try this transformation on the hazard rate 1nhtl 111W Hula t 5 11ntl 5 This is not linear in alpha and beta but it is close if we do not try and separately identify those two parameters and let the intercept equal ln llnoc and the slope equal beta minus one ie lnht a b lnt 6 We are interested in the slope If it is not signi cantly different from zero then we can accept the null hypothesis that beta equals one The logarithm of the hazard rate is plotted against the logarithm of survival time duration in Figurel Oct 31 2008 LEC 11 ECON 240A3 L Phillips Weibull Distribution Transformations Poisson Distribution Figure 1 Log Hazard Rate Vs Log Duration y 0366x 51181 R2 00549 Log H azard Rate Log Duration The coefficient of determination is only 55 The Fdistribution statistic calculated from this R2 is not signi cant and Student s tstatistic for the null hypothesis that the slope is zero is only 064 so we can not reject the hypothesis that beta equals one and the hazard rate is constant Of course there are only a few observations but the exponential seems appropriate IV Cumulative Hazard Function There is another test for whether the exponential distribution is appropriate for the duration of postwar expansions It is the cumulative hazard function Ht which is the sum of the hazard function ht Ht j hu du 7 0 Oct 31 2008 LEC 11 ECON 240A4 L Phillips Weibull Distribution Transformations Poisson Distribution And applying this to the exponential where from Eq 20 of chapter ten the hazard rate is the constant lambda t t HtJ huduJ xdum 8 0 0 So the cumulative hazard function for the exponential is a linear function of the time until failure We can calculate the cumulative hazard rate using the values we calculated for the interval hazard rate in Table 5 of the previous chapter reproduced in part in Table 1 below The cumulative hazard rate is just the running sum of the hazard rate so at Table 1 Estimated Hazard Rate and Cumulative Hazard Rate PostWar Expansions Duration Ending at Risk Interval Hazard Rate Cumulative Hazard Rate 0 0 10 12 1 10 01000 01000 24 1 9 01111 02111 36 1 8 01250 03361 37 1 7 01429 04790 39 1 6 01667 06456 45 1 5 02000 08456 58 1 4 02500 10956 92 1 3 03333 14290 106 1 2 05000 19290 125 1 Oct 31 2008 LEC 11 ECON 240A5 L Phillips Weibull Distribution Transformations Poisson Distribution the expansion of duration 12 months the cumulative hazard rate equals the hazard rate at the expansion of duration 24 months the cumulative hazard rate is equal to the hazard rate at that duration plus the previous hazard rate ie 0 l l 11 010000 and so on This cumulative hazard rate is plotted against duration in Figure 2 Figure 2 Cumulative Hazard Plot for Expansions 25 2 o y 00192X 01736 R2 09562 39o 6 N 15 g o 2 16 S 9 g 1 o o 0 Cumulative Hazard Function Linear Regression O 05 o o o 0 i i i i i 0 20 40 60 80 100 120 Duration in Nonths Note that the linear regression is a good t with an R2 of 096 supporting evidence that the expansion durations are distributed exponentially Note also that the slope which is an estimate of the hazard rate or lambda is 00192 Since the reciprocal of lambda is the mean time to failure we get an estimate of 52 months So the cumulative hazard function is quite informative and we can learn a lot using exploratory graphical analysis Oct 31 2008 LEC 11 ECON 240A6 L Phillips Weibull Distribution Transformations Poisson Distribution V Poisson Distribution The Poisson is a useful approximation to the binomial for events with a low probability ie p small one minus p or q approaching one and a sample size n of fty or more or the product of n and p equal to ve or less The density for the Poisson is fX eXpH uquot x 9 The assumptions underlying the use of the Poisson to events in intervals of space or time are l the number of events occurring in nonoverlapping intervals are independent 2 the probability of a single event occurring in a small interval is approximately proportional to the interval and 3 the probability of more than one event occurring in an interval is negligible An example of the application of the Poisson follows Ten percent of the tools produced in a manufacturing process turn out to be defective What is the probability of nding exactly two tools that are defective in a random sample of ten tools First applying the binomial n10 p0 l and npl pk2 nnkk pkqquot39k 1082012098 01937 10 as an alternative apply the Poisson where its mean u equals np pk2 expp uk k expl l2 2 12le 0184 11 where expl e 2718 Another example of application of the Poisson follows Experience has shown that the mean number of telephone calls arriving at a switchboard during one minute is ve If the switchboard can handle a maximum of eight calls per minute what is the probability it will not be able to handle all the calls arriving in a minute forcing 422009 Lecture Two 1 l Inertial Versus Causal Models A causal model is based on economic theory and can be expressed in the form of quotstructuralquot or alternatively as reduced form equations An example might be a demand and supply model for the price of gold The factors affecting demand might include the rate of in ation in the consumer price index AM and an index of world political stability s as well as income y and price p qdepfygAllhsu1y where d e f g and h are parameters and U1 is an error term Supply is given by the aggregate curve for all producers oftheir marginal cost of production mc which in turn depends upon the quantity supplied q and an index of real wages w mc a bq cw u where a b and c are parameters and U2 is an error term The quantity demanded equals the quantity supplied at the value where price equals marginal cost mcp Quantity and price are jointly determined by these quotstructuralquot demand and supply curves These equations can be solved to express these endogenous variables in terms ofthe exogenous variables For example the expression or reduced form for quantity is q 11 eb d ea ecw fy gAl hs U1 6U2 In order to estimate structural equations it is necessary to be sufficiently knowledgeable about the causal factors to correctly specify the equations It is also necessary to have access to the data describing the behavior of the exogenous variables An alternative approach is to specify an inertial model sometimes referred to in time series analysis as a structural model not to be confused with terminology used for structural causal models A simple example ofan inertial model forthe price of gold might be a time trend t and an error term u pt at Mt ut Notice that the value of price in the previous period is 422009 Lecture Two 2 pt1 at 5 t1 ut1 and the difference in price Ap de ned as APP PWU APHU U0 i e differencing removes the trend The simple time trend model requires much less information or data to estimate but is less informative than the structural model Thus there is a tradeoff Inertial models are often useful if the investigator is limited by data andor time ll Structurallnertial Models of Time Series A simple concept underlying the structural approach to time series is to conceive of the time series as composed of components 1 trend 2 cyclical 3 seasonal and 4 irregular or residual A simple additive formulation would be time seriest trendt cyclet seasonalt irregulart One could construct the time series as the sum ofthese components Le a synthesis ofthe components Of course one needs to model each component A simple model ofthe price of gold was used in the example above where price was the sum of a linear trend plus an irregular or error term Much ofthis course will be concerned with developing alternative approaches to modeling these components Often the only data the investigator has is the time series itself From the time series one can try and develop representations ofthe components This is called decomposition the inverse of synthesis A simple example ofthe concept involved can be illustrated with a back of the envelope technique using a monthly time series forthe unemployment rate and a BuysBallot table The latter is simply an array of the data with month heading the columns and years heading the rows An average for a row is the annual average and is a measure of the two components cycle plus trend having averaged out the seasonal and the irregular An average for a column is a measure ofthe seasonal component having averaged over trend cycle and irregular Thus we have created representations of some of the components or combinations of components from the time series itself an example of decomposition using a BuysBallot table 422009 Lecture Two 3 Civilian Unemployment Rate Year Jan Feb Mar Apr May Jun Jul Agg Sep Oct Nov Dec AA 82 86 89 90 93 94 96 98 98 101 104 108 108 97 83 104 104 103 102 101 101 94 95 92 88 85 83 96 84 80 78 78 77 74 72 75 75 73 74 72 73 75 85 74 72 72 73 72 73 74 71 71 72 70 70 72 86 67 72 71 72 72 72 70 69 70 70 69 67 70 87 66 66 65 64 63 62 61 60 59 60 59 58 62 88 58 57 56 55 56 54 54 56 54 53 54 53 55 89 54 51 50 53 52 53 53 53 53 53 53 53 53 90 53 53 53 54 53 53 55 56 57 57 59 61 55 91 63 65 68 66 68 68 67 68 67 68 69 72 67 92 71 74 73 73 75 77 75 75 75 73 73 73 74 93 71 70 70 70 69 69 68 67 67 67 65 64 68 94 67 66 65 64 61 61 61 60 58 57 56 54 61 Av 70 71 70 71 70 70 70 70 69 69 69 68 70 Source United States Department of Commerce Business Conditions Digest and Survey of Current Business 422009 Lecture Two 4 Annual Average ofthe Civilian Unerrployment Rate Percent Year Civilian Une no nt Rate MonthlyAverage for Thirteen Years Civilian Unemployment Rate Percent Month Avera e 422009 Lecture Two 5 III Deterministic and Stochastic Time Series A deterministic time series is a sequence ofvalues indexed by a linear index in this case time t where the values are not random An example is y 2t Deterministic Series Y m Time Y 8 2 4 o o 1 2 3 4 3 6 Time 4 8 This example ofa trended deterministic time series is growing or evolving and hence is called evolutionary It is also easily forecastable A Random Variables and Stochastic Sequences A continuous random variable x takes on values in some range with likelihood determined by its density function fx An example of such a density function is the uniform with fx 1 for x taking values in the range zero to one ie 05x51The distribution function Fx is the integral ofthe density function Fx fxdx 0 422009 Lecture Two 6 and for the uniform distribution the distribution function is FX X Density ofthe uniform distribution Distribution Function ofthe Uniform FX The expected value of a random variable is co EX f fXX dX 0 which for a variable distributed uniformly on the interval zero to one is one half The variance of a random variable is 422009 Lecture Two 7 EX EX21 which in the case of the uniform variable above is 112 A uniformly distributed random variable can be simulated using EViews File Menu New New Object V ndow Workfile Uniform Work le Range Undated Start Observations 1 End Observations 10 Work le Menu GENR Equation xRND Work le Select Object X Work le Menu View Open Selection Series X Window View Spreadsheet Series X Window View Line Graph Series X Window View Histogram and Stats The command Spreadsheet lists the ten independent and uniformly distributed observations the rst ve in the rst row followed by the second five in the next row 1 2 3 4 5 Modi ed 1 10 xrnd 0627857 0631123 0738304 0912626 0509690 0077609 0224799 0052492 0707297 0866146 Ten Observations from the Uniform Distribution The command Histogram and Stats can be used to obtain descriptive statistics for these ten observations on the uniform variable Histogram and statistics for 10 observations from the Uniform distribution 4 22009 Lecture Two 8 6 Series X Sample 1 10 539 Observations 10 I 4 39 39I Mean 0534794 I Median 0629490 3 I 39 Maximum 0912626 39 1 1 39 39 Minimum 0052492 I II Std Dev 0312699 2 I Skewness 0 479597 39u 39 39l 39u 39l 39 Kurtosis 1 652887 1 39 I39 I39 If I39 JarqueBera 1139486 0 39u 39u 39 39u 39u 39 Probability 0565671 000 025 050 0 5 1 00 Note that the sample mean of 0535 is close to the expectation of05 and that the standard deviation of 03127 implies a sample variance of0098 not too far from the expectation of 0083 Note also that the minimum and maximum observations fall within the expected range The sample often observations generated from the uniform distribution is a sequence identically distributed and independent ofone another ie the second observation is independent of the rst and third observations etc If we index these observations by time we have a stochastic time series The command Line Graph produces the trace of the time series 422009 Lecture Two 9 08 06 04 02 00 6 Trace of the time series of ten observations Uniform distribution This time series stays within the bounds ofzero and one and hence is stationary not evolutionary A series of 100 observations can be generated in a similar fashion Using the command Histogram and Stats this sample39s density function can be illustrated Histogram and statistics 100 observations Uniform distribution 422009 Lecture Two 10 10 Series X Sample 1 100 8 Observations 100 Mean 0469755 6 Median 0469832 Maximum 0980651 Minimum 0020844 4 Std Dev 0258724 Skewness 0071 820 Kurtosis 1 955055 2 JarqueBera 4635594 Probability 0098490 n IV Stationary Stochastic Process Recall the generation of 100 observations from the uniform distribution If we consider this a sample at a point in time the order ofthe observations does not matter However if we index these observations by time in the order they were drawn this time index xes the order and we have a sequence of observations or time series The trace of this time series follows 422009 Lecture Two 11 10 08 06 04 02 00 1o 20 3 4o 5dquot39639dquot397390quot39839dquot90 3913900 Uniform distribution time series By design and construction this time series is strictly stationary The first fifty observations are a sequence of identically distributed observations independent from one another The same can be said ofthe second fty observations 80 the halves of this series are conceptually indistinguishable This expectation is borne out in practice by comparing the descriptive characteristics ofthe two samples which provide estimates ofthe parameters ofthe parent uniform distribution The Sample command in EViews can be used to select the first fifty 422009 Lecture Two 12 observations and the Histogram and Stats command yields the following descriptive statistics for this sample 52 Series X Sample 1 50 Observations 50 6 Mean 0486257 Median 0501053 Maximum 0961211 439 Minimum 0020844 Std Dev 0267334 Skewness 01 14270 2 Kurtosis 1852169 JarqueBera 2853636 Probability 0240072 n 0390 0392 0394 39 39 0396 0398 1390 Similarly the second sample can be selected Sample 51 100 and the Histogram and Stats command generates the descriptive statistics for this sample 4152008 Lecture Five 1 First Order Autoregressive Processes First order autoregressive processes AR1 have a structure similar to random walks ARt bARt1 et Ifthe coef cient b equals one we have a random walk lfb is greaterthan one the process is also evolutionary If b is positive and less than one there is a positive dependence on the past but the process is stationary Many economic series are characterized with a positive autoregressive structure If b equals zero we have a process oforder zero ie white noise If b is between minus one and zero the process depends negatively on the past but is covariance stationary If b is algebraically less than one the process is evolutionary The following plot shows three evolutionary autoregressive processes for values of b of 1 5 1 and 15 respectively Th39ee Evoluiona39y F39rst OderAJtoregressive Fl39ccesses ARP15ARP 1WN ARN 15ARN 1 WN RWRW1WN TINE PRN PRP 39 Using the lag operator an autoregressive process can be expressed as 4152008 Lecture Five 2 1 bZ ARt et so that 1bZ is the lter that converts a first order autoregressive process to white noise Multiplying both sides by the inverse ofthis lter 1bZ11bZ ARt ARt 1bZ1et or ARt 1 bZ bZZ2 b323 et et bZet b222et ARt et bet1 bZetZ b3et3 The mean function ofthis autoregressive process is mt EARt Eet bEet1 b2Eet2 b3Eet3 O and the autocovariance function at lag zero is ARVm Eet bet1 bZet2 et bet1 bZetZ 62 b262 b464 621b2 If we lag the autoregressive process by one period ARt1 et1 bet2 bZet3 b3et4 we see that it depends upon past values of the white noise series and if the parameter b is less than one in absolute value the influence ofthe past will fade The autocorrelation at lag one can be calculated from YARAR1 EARt ARC1 EARt1 bARt1 et bEARt1 ARt1 b YARAR0 so that the autocorrelation function at lag one is b PARAR1 b Recall from above that ARt1 did not depend upon et so that the covariance of ARt1 and et is zero Similarly the autocovariance at lag two is 4152008 Lecture Five 3 YARAR2 EARt ARt2 EARt2 bARt1 et bEARt2 ARt1 bEARt1 ARt b YARAR1 b2 ARAR0 and the autocorrelation function at lag two is b2 PARAR2 b2 and in general PARARu b for all u20 so that the autocorrelation function will decay with lag Two examples for the parameter b equal to positive and negative 09 are illustrated below First Order Autoreg ressiv e 39 b09 D b09 2 VACmare LAG 05 Autocorrelation 1 An autoregressive process with a positive coefficient approaching one will have an autocorrelation function that is dif cult to distinguish from the autocorrelation function ofa random walk sample or a trended sample 4152008 Lecture Five 4 If we use Monte Carlo techniques to create an autoregressive process of 100 observations from a white noise process we can use an approach similar to the one we used to generate a random walk We set the AR process for the rst observation equal to the rst observation ofwhite noise and then use the following expression for the second observation AR2 07 AR1 WN2 and so forth for the remaining observations The trace of such a simulated process is plotted Simulated First Order Autoregressive Process AR ON Et 07 ARON Et1 WNt TIME The estimated autocorrelation function for this simulated sample is displayed at the top of page 5 As we derived above the value of the autocorrelation function at lag one is an estimate of the parameter b for the rst order autoregressive process Because of the relative small sample size the estimate of0463 is considerably below the expected value of07 The model can also be estimated using the command menu in TSP and choosing the time series option and then selecting ARMA estimation and using the LS command LS R ARONE CARONE1 we obtain the model estimates displayed on the bottom of this page 4152008 Lecture Five 5 IDENT ARONE SMPL range 1 100 Number of observations 100 Autocorrelations PartialAutocorrelations ac pac cum I cum I 1 0463 0463 2 0201 0016 3 0122 0045 4 0046 0032 5 0050 0045 600100064 700960091 801680113 903200241 10 0229 0030 QStatistic 10 lags 46685 SE of Correlations 0100 LS Dependent Variable is ARONE SMPL range 2 100 Sample endpoints adjusted to exclude missing data Number of observations 99 VARIABLE COEFFICIENT STD ERROR TSTAT 2TAIL SIG C 01670034 01082534 15427088 0127 ARONE1 04633409 00897108 51648303 0000 Rsquared 0215689 Mean of dependent 0315322 Adjusted Rsquared 0207604 SD of dependent 1166655 SE ofregression 1038517 Sum of squared resid 1046163 DurbinWatson stat 1993228 Fstatistic 2667547 Log likelihood 1432063 Note that the estimate of the coefficient on ARONE1 is the same as we obtained from the estimated autocorrelation function A plot ofthe simulated observations the values tted from the estimated model and the estimated residuals are shown below 4152008 Lecture Five 6 First Order Autoregressive Smulation b 07 Series Fitted amp Residuals 20 40 60 80 1 00 RESIDUAL ARONE FI39I39I39E The autocorrelogram of the residuals is displayed on page 8 These residuals look white A histogram of the residuals reveals an approximately bell shaped distribution One could go on and calculate their observed distribution function to check further for the expected normality A scatter plot of the estimated residuals against the observations of the simulated white noise series shows that the rst order autoregressive model approximately recovers the original white noise input 4152008 Lecture Five 7 Histogram of Res39dualsfrom the First Order Autoregressive Model b07 Scatter Plot of Residuals from AR1 Model b07 Against White Noise Input 3 RES Lecture Eleven 1 572009 Dynamic Causal Models Our discussion of dynamic relationships between two or more variables will be conducted within the conceptual and historical framework of permanent consumption The idea is that permanent consumption cp is proportional to permanent income yp which in turn is the return on your human and other property and nancial wealth cp k yp Observed consumption 0 is conceived to be the sum of permanent consumption cp and transitory consumption cT where the latter is assumed to be white noise c cp cT Combining these two relationships we have c k yp CT An historical approach to formulating permanent income within a time series context was as a distributed lag of current and past observed income ypt ho Yt M Yt1 Mild2 lava3 So the complete model would be c k Mya k M yt1 k x2yt 2 k A3yt3 cT Econometric Considerations One could estimate this distributed lag model of consumption on income by regressing current consumption on current and lagged values of income using multiple linear regression However we can anticipate that income will have an autoregressive structure and hence that current income will be highly correlated with lagged income etc Thus the explanatory variables yt yt1 yt2 will be multicollinear which can lead to large standard errorsand low tstatistics for the estimated parameters k we k M k x2 etc Possible econometric alternatives to estimation have included the Almon polynomial distributed lagPDL technique imposition ofa lag Lecture Eleven 2 572009 structure such as the geometric Koyck differencing both sides ofthe equation before estimation and the BoxJenkins approach to estimating distributed lags The rst step of the Almon technique is to choose the length of the distributed lag In our example we will end the lagged dependence ortruncate it after three lags ie assume that k M is zero and likewise for the coefficients on higher lags The second step is to assume that we will approximate the lag structure k we k M k A2 etc with a polynomial For example suppose we choose a polynomial oforder one a b i where i indexes lag number Then for the lag structure we have kxo a kw a b km a 2b kx3 a 3b and substituting in the distributed lag model we have from c k ho Yt k A1Yt1 k 7L2 Yt2 k M Yt3 cT C aya abYt1 32bYt2 33bYt3 CT and collecting the variables for the coefficients a and b C allD Yt1 Yt2 Yt3 blYt1 2Yt2 3Yt3 CT The variables yt yt1 yt2 yt3 and yt1 2yt2 3yt3 are referred to as the Almon variables The lag structure can be recovered from the estimates of a and b ie k we a kx1a betc A third approach is to difference the distributed lag model noting that differencing income will prewhiten it and hence reduce collinearity between Ayt Ayt1 etc The differenced model is Ac k we Ayt k M Ayt1 k M A yt2 k M Ayt3 ACT lfone encounters multicollinearity in the attempt to estimate the model in levels this offers a simple alternative However ifthe structure in the explanatory variable is more complex than Lecture Eleven 3 572009 AR1 with a parameter close to one then rst differencing may not be a good approximation This motivates the more general BoxJenkins approach Before discussing the BoxJenkins technique for estimating distributed lags it is important to mention another approach the imposition ofa lag structure such as geometric For example suppose or in general Ki 9 ri and substituting in the distributed lag model c keyt keryt1 ker2yt2 ker3yt3 cTY c ke1 rZ r222 r323 yt 0139 c ke1rZ yt cT Multiplying through by 1rZ ct rct1 keyt cTt rcTt1 Estimation now involves a lagged dependent variable Ct rCt1 kGYG 0T0 re t1 Note also that the error stucture cTt rcTt1 is MA1 The econometric dif culty is that the lagged dependent variable ct1 is not independent ofthis error structure cTt rcTt1 and hence the parameter estimates r and k6 will be biased and inconsistent Since transitory consumption cTt is the random or error component ofobserved Lecture Eleven 4 572009 consumption ct it follows that ct1 the lagged dependent variable will be correlated with cTt 1 and hence not independent ofthe error structure violating the assumptions for linear regression BoxJenkins Estimation of Dynamic Distributed Lag Models For a dynamic relationship ofthe form yt h0xt h1xt1 h2xt2 ut where Xt is the input yt is the output and ut is the error structure Using the lag operator yt hOZ h122 h223 xt ut where hOZ h122 h223 is the lter or impulse response function that transforms the input into the signal component ofthe output The rest ofthe output is noise This lter can be written in compact form as a polynomial in The lag operator Z hZ hZ hOZ h1Z2 hZZ3 so that Yt hZ Xt ut The econometric problem is how to estimate hZ lfxt is ARMApq ie ARP Xt MAM ext or equivalently 1b1Zb222prP Xt 1 a1Za222anq ext mutiplying both sides of the inputoutput relationship by the ratio ARpMAq will reduce the input Xt to its white noise residual ARpMAqyt hZ ARpMAqXt ARpMAqut or equivalently ARPMAQYt hZ ext ARPMAQUt Lectures on Differential Geometry Math 24OBC John Douglas Moore Department of Mathematics University of California Santa Barbara7 CA7 USA 93106 e mail moore mathuosbedu June 57 2009 Preface This is a set of lecture notes for the course Math 240BC given during the Winter and Spring of 2009 The notes evolved as the course progressed and are still somewhat rough but we hope they are helpful Starred sections represent digressions are less central to the core subject matter of the course and can be omitted on a first reading Our goal was to present the key ideas of Riemannian geometry up to the generalized GaussBonnet Theoremi The first chapter provides the foundational results for Riemannian geometry The second chapter provides an introduction to de Rham cohomology which provides prehaps the simplest introduction to the notion of homology and cohomology that is so pervasive in modern geometry and topology In the third chapter we provide some of the basic theorem relating the curvature to the topology of a Riemannian manifoldithe idea here is to develop some intuition for curvaturei Finally in the fourth chapter we describe Cartanls method of moving frames and focus on its application to one of the key theorems in Riemannian geometry the generalized GaussBonnet Theoremi The last chapter is more advanced in nature and not usually treated in the firstyear differential geometry course It provides an introduction to the theory of characteristic classes explaining how these could be generated by looking for extensions of the generalized Gauss Bonnet Theorem and describes applications of characteristic classes to the Atiyah Singer Index Theorem and to the existence of exotic differentiable structures on sevenspheres Contents 1 Riemannian geometry 2 lil Review of tangent and cotangent spaces i i i i i i i i i i i i i i 2 1 2 Riemannian metrics i i i i i i i i i i i i i i i i i i i i i i i i i 5 ll Geodesics i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 9 131 Smooth paths i i i i i i i i i i i i i i i i i i i i i i i i i 9 132 Piecewise smooth paths i i i i i i i i i i i i i i i i i i i 13 1 4 Hamilton s principle i i i i i i i i i i i i i i i i i i i i i i i i i 14 15 The LeviCivita connection i i i i i i i i i i i i i i i i i i i i i 20 16 First variation 0 i i i i i i i i i i i i i i i i i i i i i i i i i i 24 1 7 Lorentz manifolds i i i i i i i i i i i i i i i i i i i i i i i i i i i 27 18 The RiemannChristoffel curvature tensor i i i i i i i i i i i i i 30 1 9 Curvature symInetries sectional curvature i i i i i i i i i i i i i 36 1 10 Gaussian curvature of surfaces i i i i i i i i i i i i i i i i i i i 39 1 11 Review of Lie groups i i i i i i i i i i i i i i i i i i i i i i i i i 42 1 12 Lie groups with biinvariant metrics i i i i i i i i i i i i i i i i i 45 1 13 Grassmann manifolds i i i i i i i i i i i i i i i i i i i i i i i i 49 114 The exponential map i i i i i i i i i i i i i i i i i i i i i i i i i 53 115 The Gauss Lemma i i i i i i i i i i i i i i i i i i i i i i i i i i 56 1 16 Curvature in normal coordinates i i i i i i i i i i i i i i i i i i 57 1 17 Riemannian manifolds as metric spaces i i i i i i i i i i i i i i i 60 1 18 Completeness i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 62 1 19 Smooth closed geodesics i i i i i i i i i i i i i i i i i i i i i i i 64 2 Differential forms 68 i Tensor algebra i i i i i i i i i i i i i i i i i i i i i i i i i i i i 68 22 The exterior derivative i i i i i i i i i i i i i i i i i i i i i i i i 71 2 3 Integration of differential forms i i i i i i i i i i i i i i i i i i i 75 24 Theorem of Sto es i i i i i i i i i i i i i i i i i i i i i i i i i i 80 25 de Rham Cohomology i i i i i i i i i i i i i i i i i i i i i i i i 83 2 6 Poincare LemIna i i i i i i i i i i i i i i i i i i i i i i i i i i i 85 2 7 MayerVietoris Sequence i i i i i i i i i i i i i i i i i i i i i i i 90 2 8 Singular homology i i i i i i i i i i i i i i i i i i i i i i i i i i 94 281 De nition of singular homology i i i i i i i i i i i i i i 94 2 82 Singular cohomology i i i i i i i i i i i i i i i i i i i i 98 CO 1 2813 Proof of the de Rham Theorem 1 1 1 1 1 1 1 1 1 1 1 1 1 1 102 219 The Hodge star 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 106 2110 The Hodge Laplacian 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 112 2111 The Hodge Theorem 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 116 2112 d and 6 in terms of moving frames 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 119 2113 The rough Laplacian 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 122 2114 The Weitzenbock formula 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 123 2115 Ricci curvature and Hodge theory 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 125 2116 The curvature operator and Hodge theory 1 1 1 1 1 1 1 1 1 1 1 1 1 126 2117 Proof of GallotMeyer Theorem 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 128 Curvature and topology 131 311 The Hadamard Cartan Theorem 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 131 32 Parallel transport along curves 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 133 313 Geodesics and curvature 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 134 314 Proof of the Hadamard Cartan Theorem 1 1 1 1 1 1 1 1 1 1 1 1 1 1 138 315 The fundamental group 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 140 3511 De nition of the fundamental group 1 1 1 1 1 1 1 1 1 1 1 1 140 31512 Homotopy lifting 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 142 31513 Universal covers 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 145 36 Uniqueness of simply connected space forms 1 1 1 1 1 1 1 1 1 1 1 1 147 37 Non simply connected space forms 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 149 318 Second variation of action 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 151 319 Myers7 Theorem 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 153 3110 Syngels Theorem 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 155 Cartan s method of moving frames 159 411 An easy method for calculating curvature 1 1 1 1 1 1 1 1 1 1 1 1 1 159 412 The curvature of a sur ace 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 163 413 The GaussBonnet formula for surfaces 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 167 414 Application to hyperbolic geometry 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 171 45 Vector bundles 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 177 46 Connections on vector bundles 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 179 417 Metric connections 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 183 418 Curvature of connections 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 185 419 The pullback construction 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 187 4110 Classi cation of connections in complex line bundles 1 1 1 1 1 1 1 189 4111 Classi cation of U1bundles 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 195 4112 The Pfa ian 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 196 4113 The generalized Gauss Bonnet Theorem 1 1 1 1 1 1 1 1 1 1 1 1 1 1 197 4114 Proof of the generalized GaussBonnet Theorem 1 1 1 1 1 1 1 1 1 200 5 Characteristic Classes 208 51 The Chern character i i i i i i i i i i i i i i i i i i i i i i i i i 208 5 2 Chern classes i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 213 53 Examples of Chern classes i l i l l i l i i i i i i i i i i l i l l i 215 5 4 Invariant polynomials i i i i i i i i i i i i i i i i i i i i i i i i 217 55 The universal bundle i i i i i i i i i i i i i i i i i i i i i i i i 220 56 The Clifford algebra i i i i i i i i i i i i i i i i i i i i i i i i i 225 57 The spin group i i i i i i i i i i i i i i i i i i i i i i i i i i i i 232 5 8 Spin structures and spin connections i i i i i i i i i i i i i i i i 235 59 The Dirac operator i i i i i i i i i i i i i i i i i i i i i i i i i i 240 510 The Atiyah Singer Index Theorem i l i i i i i i i i i i l i l l i 243 5101 Index of the Dirac operator i i i i i i i i i i i i i i i i i 243 5102 SpinC structures i i i i i i i i i i i i i i i i i i i i i i i 246 5103 Dirac operators on general manifolds i i i i i i i i i i i 248 5104 Topological invariants of fourmanifolds i i i i i i i i i i 252 511 Exotic spheres i i i i i i i i i i i i i i i i i i i i i i i i i i i i 253 Bibliography 257 Chapter 1 Riemannian geometry 11 Review of tangent and cotangent spaces We Will assume some familiarity With the theory of smooth manifolds as pre sented for example in the rst four chapters of Suppose that M is a smooth manifold and p E M and that fp denotes the space of pairs U Where U is an open subset of M containing p and f U is a smooth function If 45 111 i 1 i U A R is a smooth coordinate system on M With p E U and U E fp we de ne a 81139 f Dif o 1 p 6 R Where Di denotes differentiation With respect to the ith component We thereby obtain an Rlinear map INQAR i 81139 p called a directional derivative operator Which satis es the Leibniz rule 6 6 f9 lt61 fgt 910 lt81 ygt 7 17 17 17 and in addition depends only on the germ of f at p E f i i p 191i 8 f E g on some neighborhood of p 67 z a 81139 a The set of all linear combinations of these basis vectors comprises the tangent space to M at p and is denoted by TpMi Thus for any given smooth coordinate system 117 i i i 7 I on M7 we have a corresponding basis lt a gt 1 81 p for the tangent space TpMi The notation we have adopted makes it easy to see how the components ai of a tangent vector transform under change of coordinates If 1 y 7 i i i 7y is a second smooth coordinate system on M7 the new basis vectors are related to the old by the chain rule7 8 n 8N 8 Byi p Byi p 11 8 7Hi77 81 p 37 p DM 0 wlgtltwltpgtgti 7 where 17 The disjoint union of all of the tangent spaces forms the tangent bundle TM UTpM p e M7 which has a projection 7r TM A M de ned by 7rTpM pi lfqb 117 i i 7 I is a coordinate system on U C M7 we can de ne a corresponding coordinate system 45 zl7ui7zn7z39l7iu7z39n on W71U C TM by letting n V a n V a z j 7 z z j 7 z z a 61 7 z p7 z a 61 7 a i 11 11 17 11 12 For the various choices of charts U7 45 the corresponding charts n 1U7q5 form an atlas making TM into a smooth manifold of dimension 2n7 as you saw in Math 240Ai The cotangent space to M at p is simply the dual space TM to TpMi Thus an element of TM is simply a linear map aszMaRi Corresponding to the basis of TpM is the dual basis n i 8 dzl pp 7dr 17 7 de ned by dz 1 gt 7 17 ifz j7 5j I p 07 1fz7 ji The elements of Tsz called cotangent vectors are just the linear combinations of these basis vectors n E aidzi lp i1 Once again under change of coordinates the basis elements transform by the chain rule Byi l 81 dyilp WWIle j An important example of cotangent vector is the differential of a function at a point If p E U and f U A R is a smooth function then the dz erential of f at p is the element dflp E TSM de ned by dflpv If 11 i i 1 is a smooth coordinate system de ned on U then n 6f dflp 61139 i1 mdxilp Just as we did for tangent spaces we can take the disjoint union of all of the cotangent spaces forms the cotangent bundle T M UTM p e M which has a projection 7r TM 4 M de ned by 7rTpM pl 1qu 11 i i I is a coordinate system on U C M we can de ne a corresponding coordinate system lt131171 7p17m7pn on 7r 1U C TM by letting n n 11 E a 76 j zlp pi E ajdzjlp aii j1 I 17 jl For the various choices of charts U 45 the corresponding charts n lUq5 form an atlas making T M into a smooth manifold of dimension n We can generalize this construction and consider tensor products of tangent and cotangent spaces For example the tensor product of the cotangent space with itself denoted by 2TM is the linear space of bilinear maps ngpMXTpMaR If 45 11 i i i 1 U gt R is a smooth coordinate system on M with p E U we can de ne drill dzjlp TpM gtlt TpM A R by dzllp dzjlp 17 Then I V dzllpcadzllp 1 g 239 g nl gjg n is a basis for 2TM and a typical element of 2T5M can be written as Z 9ijltPgtdIilp delm Zj where the gij p7s are elements of R 12 Riemannian metrics De nition Let M be a smooth manifold A Riemannian metn39c on M is a function which assigns to each p E M a positivede nite inner product lt gtp on TpM which varies smoothly77 with p 6 Mi A Riemannian manifold is a pair M lt consisting of a smooth manifold M together with a Riemannian metric lt on Of course we have to explain what we mean by vary smoothlyi77 This is most easily done in terms of local coordinates If 45 11 i i 1 U A R is a smooth coordinate system on M then for each choice of p E U we can write ltv 39gtp Z 9ijlt10gtd1ilp drjlp 1171 We thus obtain functions gij U A K To say that lt gtp varies smoothly with p simply means that the functions gij are smooth We call the functions gij the components of the metric Note that the functions gij satisfy the symmetry condition gij gji and the condition that the matrix gij be positive de nite We will sometimes write n lt Z gijdzi dzji ij1 If 1 yl i i y is a second smooth coordinate system on V g M with n lt3 39gtlV Z hijdyl dij ij1 it follows from the chain rule that on U N V n Byk Byl 91739 Z hklaziw 1611 We will sometimes adopt the Einstein summation convention and leave out the summation sign Byk Byl We remark in passing that this is how a covariant tensor eld of rank two transforms under change of coordinates Using a Riemannian metric one can lower the index of a tangent vector at p producing a corresponding cotangent vector and vice versa Indeed if v E TpM we can construct a corresponding cotangent vector av by the formla gij hkl avltwgt ltv wgtpv In terms of components n if v Zai all then av Z gijpajdzilpi i1 17 i1 Similarly given a cotangent vector 1 E TM we raise the index to obtain a corresponding tangent vector va 6 TpMi In terms of components a 81139 7 Z7 n n if a Zaidzilp then v0 Z gijpaj 391 1 z where 917 is the matrix inverse to gij Thus a Riemannian metric transforms the differential dflp of a function to a tangent vector gradfp g 010 v 17 called the gradient of f at p Needless to say in elementary several variable calculus this raising and lowering of indices is done all the time using the usual Euclidean dot product as Riemannian metric Example 1 Indeed the simplest example of a Riemannian manifold is n dimensional Euclidean space E which is simply R together with its standard rectangular cartesian coordinate system 111 1 1 I and the Euclidean metric lt dzl dzldzn dzni In this case the components of the metric are simply 1 ifz39 j g 6 0 ifz ye We will often think of the Euclidean metric as being de ned by the dot product n ia n V a n ia n V a n H ltZa HZ 11 17 17 11 17 11 17 j1 i1 Example 2 Suppose that M is an ndimensional smooth manifold and that F M A RN is a smooth imbedding We can give RN the Euclidean metric de ned in the preceding example For each choice of p E M we can then de ne an inner product lt gtp on TpM y ltvwgtp Fpv Fpw for 1110 6 Tle Here Fig7 is the differential of F at p de ned in terms of a smooth coordinate system 45 11 l l l I by the explicit formula 6 Clearly ltvwgtp is symmetric and it is positive de nite because F is an immer sionl Moreover 8 8 Fpltii gtIFWlt7V pgt 81 p 81 p 9M lt DiF 0 1 P 39DjF 0 l 107 so gij p depends smoothly on p Thus the imbedding F induces a Riemannian metric lt on M which we call the induced metric and we write lt3 Flt397 gt BM 0 ar1gtlt ltpgtgt 6 RN 7 J p 81 It is an interesting fact that this construction includes all Riemannian manifolds De nition Let M lt be a Riemannian manifold and suppose that EN denotes RN with the Euclidean metric An imbedding F M A EN is said to be isometric if lt Flt Nash s Imbedding Theorem If M lt is any smooth Riemaunian man ifold there exists an isometric imbedding F 4 EN into some Euclidean space This was regarded as a landmark theorem when it rst appeared 28 The proof is dif cult involves subtle techniques from the theory of nonlinear partial differential equations and is beyond the scope of this course special case of Example 2 consists of twodimensional smooth manifolds which are imbedded in Egl These are usually called smooth surfaces in E3 and are studied extensively in undergraduate courses in curves and surfaces77 This subject was extensively developed during the nineteenth century and was summarized in 188796 in a monumental fourvolume work Lecoris sur la theorie g ri rale des surfaces et les applications g om triques du calcul in nitesimal by Jean Gaston Darbouxl Indeed the theory of smooth surfaces in E3 still provides much geometric intuition regarding Riemannian geometry of higher dimensions What kind of geometry does a Riemannian metric provide a smooth manifold M Well to begin with we can use a Riemannian metric to de ne the lengths of tangent vectorsi If v E TpM we de ne the length of v by the formula HvH ltv7vgtpA Second we can use the Riemannian metric to de ne angles between vectors The angle 9 between two nonzero vectors 1110 6 TpM is the unique 9 E 07r such that ltv7wgtp llvllllwll COS 9 Third one can use the Riemannian metric to de ne lengths of curves Suppose that 7 z a b A M is a smooth curve with velocity vector M 3 dt 81139 11 7t E Tqu for t E a 12 yt Then the length of 7 is given by the integral 1 LW ltwlttwlttgtgtmdt We can also write this in local coordinates as b n dzi dzj LW Z gijltvlttgtgtggdt 1 ij1 Note that if F M A lEN is an isometric imbedding then LW LF o 7 Thus the lengths of a curve on a smooth surface in E3 is just the length of the corresponding curve in Egi Since any Riemannian manifold can be isometrically imbedded in some EN one might be tempted to try to study the Riemannian geometry of M via the Euclidean geometry of the ambient Euclidean space However this is not necessarily an ef cient approach since sometimes the iso metric imbedding is quite dif cult to construct Example 3 Suppose that H2 1 y E R2 z y gt 0 with Riemannian metric l lt Ear dz dy dyi A celebrated theorem of David Hilbert states that H2 lt has no isometric imbedding in E3 and although isometric imbeddings in Euclidean spaces of higher dimension can be constructed none of them is easy to describe The Riemannian manifold H2 lt is called the Poincar upper halfplane and gures prominently in the theory of Riemann surfaces 13 Geodesics Our rst goal is to generalize concepts from Euclidean geometry to Riemannian geometry One of principal concepts in Euclidean geometry is the notion of straight line What is the analog of this concept in Riemannian geometry One candidate would be the curve between two points in a Riemannian manifold which has shortest length if such a curve exists 131 Smooth paths Suppose that p and q are points in the Riemannian manifold M lt If a and b are real numbers with a lt b we let 9abMp4 Smooth paths 71 712 A MI 7a p707 q We can de ne two functions L J QabMpq A R by b b Lv ltv tw tgtzdt7 J7 ltv lttgtv lttgtgtdt Although our goal is to understand the length L it is often convenient to study this by means of the closely related action J Notice that L is invariant under reparametrization of 7 so once we nd a single curve which minimizes L we have an in nitedimensional famin This together with the fact that the formula for L contains a troublesome radical in the integrand make J far easier to work with than L It is convenient to regard J as a smooth function on the in nitedimensional manifold 9pm M p q At rst we use the notion of in nitedimensional man ifold somewhat informally but later we will return to make the notion precise Proposition 1 L72 S 21 7 aJ7 Moreover equality holds if and only if MOMO is constant if and only ify has constant speed Proof We use the CauchySchwarz inequality b 2 LW l wltwlttgtgtdtl S b b dtl l wwwwl 2ltbeagtJltvgt 12gt Equality holds if and only if the functions lt7 t7 tgt and 1 are linearly de pendent that is if and only if 7 has constant speed Proposition 2 Suppose that M has dimension at least two An element 7 E QabMp q minimizes J if and only if it minimizes L and has constant speed Sketch of proof One direction is easy Suppose that 7 has constant speed and minimizes L Then if A E QabMpq 212 a W Mn 3 L00 3 21 7 LNG and hence J7 S JAi We will only sketch the proof of the other direction for now later a complete proof will be available Suppose that 7 minimizes J but does not minimize L so there is A E 9 such that LA lt Lyi Approximate A by an immersion A1 such that LA1 lt Ly this is possible by a special case of an approximation theorem due to Whitney see 15 page 27 Since the derivative X1 is never zero the function 8t de ned by slttgt WOW is invertible and A1 can be reparametrized by arc length It follows that we can nd an element of A2 a b A M which is a reparametrization of A1 of constant speed But then 21 aJ2 LOW LA1l2 lt MW S 21 a W a contradiction since 7 was supposed to minimize J Hence 7 must in fact minimize Li By a similar argument one shows that if 7 minimizes J it must have constant speed The preceding propositions motivate use of the function J 9pm M p q A R instead of i We want to develop enough of the calculus on the in nite dimensional manifold77 QabMpq to enable us to nd the critical points of J To start with we need the notion of a smooth curve 1 766 A QabMpq such that 10 7 where y is a given element of 9 We would like to de ne smooth charts on the path space 9pm M p 4 but for now a simpler approach will suf ce We will say that a van39atz39zm of 7 is a map it 676 A 9abMp74 such that 10 7 and if a 66 gtlt ab gt M is de ned by ast dst then a is smooth De nition An element 7 E QabMpq is a critical point for J if di Jd 0 for every variation 1 of 7 13 3 50 10 De nition An element 7 E QabMp q is called a geodesic if it is a critical point for J Thus the geodesics are the candidates for curves of shortest length from p to 417 that is candidates for the notion of straight line in Riemannian geometryl We would like to be able to determine the geodesics in Riemannian manifolds It is easiest to do this for the case of a Riemannian manifold M7 lt that has been provided with an isometric imbedding in ENl Thus we imagine that M Q EN and thus each tangent space TpM can be regarded as a linear subspace of RNl Moreover7 ltvwgtp v w7 for 1110 6 TpM Q EN where the dot on the right is the dot product in ENl If it p 9abMpyq is a variation of an element 7 E QM M p q7 with corresponding map 0127676gtltalzgtM lEN7 then d 1 7 6a 6a 0 E 5a Estdt 70 b 2 E 1 8a 70 a W 39 WWW where a is regarded as an lENvalued function If we integrate by parts7 and use the fact that d E ltJltaltsgtgtgt 5 b 2 E 1 8a a fasatstEstdt 8a 8a E03717 0 gala we nd that 1 2a b ltJltaltsgtgtgtsoe ltotgtZ7ltotgtdte VOWwt lt14 where Vt Ba83 077 is called the van39atz39zm eld of the variation eld it Note that Vt can be an arbitrary smooth lENvalued function such that Va 07 V02 07 Vt E T QM for all t 6 mb that is7 V can be an arbitrary element of the tangent space77 T79 smooth maps V a7 12 A EN such that Va 0 V027 Vt E Tqu for t 6 ab 11 We can de ne a linear map dJW T79 A R by b d WWW ltVt77 tgtdt E MSW 7 a 50 for any variation 1 with variation eld Vi We think of dJW as the dz erential of J at 7 If dJW V 0 for all V E T79 then 7 t must be perpendicular to T QM for all t 6 ab In other words 7 ab A M is a geodesic if and only if 777T 0 for all t e w 15 where y tT denotes the orthogonal projection of 7 t into T QMl To see this rigorously we choose a smooth function 77 z a b gt R such that 77a 0 771 77 gt 0 on ab and set T Wt 77G WNW 7 which is clearly an element of T751 Then dJW V 0 implies that 11377lt77lttgtgtT7 lttgtdt 1377 H 777ng dt 0 Since the integrand is nonnegative it must vanish identically and 15 must indeed holdi We have thus obtained a simple equation 15 which characterizes geodesics in a submanifold M of ENl The geodesic equation is a generalization of the simplest secondorder linear ordinary differential equation the equation of a particle moving with zero acceleration in Euclidean space which asks for a vectorvalued function 7 z a b 8 EN such that 7 t 0 Its solutions are the constant speed straight lines The simplest way to make this differential equation nonlinear is to consider an imbedded submanifold M of EN with the induced Riemannian metric and ask for a function 7 ab A M C EN such that yKt T 0 In simple terms we are asking for the curves which are as straight as possible subject to the constraint that they lie within Ml Example Suppose that M S zlluzn1 E Enlz 112 z 12 1 Let el and e2 be two unitlength vectors in S which are perpendicular to each other and de ne the unitspeed great circle 7 z a b A S by 7t cos tel sin tegl 12 Then a direct calculation shows that y t 7 costel 7 sinteg 770 Hence 7 tT 0 and 7 is a geodesic on Sni We will see later that all geodesics on S are obtained in this manner 132 Piecewise smooth paths Instead of smooth paths7 we could have followed Milnor 25 117 and considered the space of piecewise smooth maps7 QabMp74 piecewise smooth paths 71 tab H M I 7a 10771 a By piecewise smooth7 we mean 7 is continuous and there exist to lt t1 lt lt tN with to a and tN I such that 7lti1ti is smooth for 1 S i S Ni In this a variation of 7 is a map it 676 A QabMp74 such that 10 7 and if a 7676 gtlt ab 4 M is de ned by ast d8t then there exist to lt t1 lt lt tN with to a and tN I such that Wise gtlt ltiilvtz l is smooth for l g i 3 Ni As before7 we nd that 1 2a a dis ltJltaltsgtgtgt 50 5650 37W but now the integration by parts is more complicated because 7 t is not con tinuous at t1 i i i tN71i If we let 7 7 7 7 ix gt331th 7 01 333V t a short calculation shows that 14 becomes d 5 N71 g 0015 0 a Wt WOW 7 Wti V OH 7 W24 5 whenever i is any variation of 7 with variation eld Vi If WWW ltJltaltsgtgtgt 70 o for all variation elds V in the tangent space T79 piecewise smooth maps V ab A EN such that Va 0 V02 Vt E Tqu for t 6 ab it follows that y ti y ti7 for every i and y tT 0 Thus critical points on the more general space of piecewise smooth paths are also smooth geodesics Exercise I Suppose that M2 is the right circular cylinder de ned by the equa tion 12 y2 1 in Egi Show that for each choice of real numbers a and b the curve cosat 7 R A M2 Q E3 de ned by 7at sinat bt is a geodesici 14 Hamilton s principle Of course we would like a formula for geodesics that does not depend upon the existence of an isometric imbeddingi To derive such a formula it is conve nient to regard the action J in a more general context namely that of classical mechanics De nition A simple mechanical system is a triple M lt gt 45 where M lt is a Riemannian manifold and 45 M A R is a smooth function We call M the con guration space of the simple mechanical systemi lf 7 ab A M represents the motion of the system 1 lt7t7tgt kinetic energy at time t 7t potential energy at time t Example 1 If a planet of mass m is moving around a star of mass M with M gtgt m the star assumed to be stationary we might take M R3 07070 lt mdz dz dy dy dz dz 7 GM m MI y 2 A 12 y2 22 Here G is the gravitational constant Sir Isaac Newton derived Kepler7s three laws from this simple mechanical systemi Example 2 To construct an interesting example in which the con guration space M is not Euclidean space we take M 503 the group of real orthogo nal 3 X 3 matrices of determinant one regarded as the space of con gurations of a rigid body B in R3 which has its center of mass located at the origin We want to describe the motion of the rigid body as a path 7 z a b A Mi If p is a point in the rigid body with coordinates 111213 at time t 0 we suppose that the coordinates of this point at time t will be 11 a11t a12t an t 70 12 7 Where 7 a2175 2205 a23t 6 503 13 a31t a32t agg t and 70 I the identity matrix Then the velocity vt of the particle p at time t will be 3 I 1 E 1 aiz W 7 t 13 El a i l v I 211 aSi ml and hence 3 w ya Z ata tziz Suppose now that pzl 12 13 is the density of matter at 11 12 13 within the rigid body Then the total kinetic energy within the rigid body at time t will be given by the expression K ltB pzl12zgzizjdzld12dzsgt aita jti We can rewrite this as K acijafki h cj t where cij A pzl12zgzizjdzld12dzg my and de ne a Riemannian metric on M 503 by WW 7 tgt Z Cijammtjw 131 Then once again l2y t 7 represents the kinetic energy this time of the rigid body B when its motion is represented by the curve 7 z a b A Mi We remark that the constants Iij Tracecij6ij 7 cij are called the moments of inertia of the rigid body A smooth function 45 503 gt R can represent the potential energy for the rigid body In classical mechanics books the motion of a top is described 15 by means of a simple mechanical system which has con guration space 503 with a suitable leftinvariant metric and potential Applied to the rotating earth the same equations explain the precession of the equinoxes according to which the axis of the earth traverses a circle in the celestial sphere once every 26000 years In Lagrangian mechanics the equations of motion for a simple mechanical sys tem are derived from a variational principle The key step is to de ne the Lagrangian to be the kinetic energy minus the potential energy More pre cisely for a simple mechanical system M lt gtq we de ne the Lagrangian L M A R by W ltv7vgt e we where 7r TM 4 M is the usual projectioni We can then de ne the action J19abMp74 HR by b M v tgtgtdt As before we say that 7 E Q is a critical point for J if 13 holds We can than formulate Lagrangian classical mechanics as follows Hamilton s principle If 7 represents the motion of a simple mechanical system then 7 is a critical point for J Thus the problem of nding curves from p to q of shortest length is put into a somewhat broader context It can be shown that if 7 E QabMpq is a critical point for J and ad Q ab then the restriction of 7 to ad is also a critical point for J this time on the space Shad M 7 s where 7 70 and s 7di Thus we can assume that ya 12 Q U where Uzli i i I is a coordinate system on an we can express L in terms of the coordinates 11 i i i I z39 i i i in on 7r1U described by 11 If 7t 11t i i i I t and MO 11tiHz ti1tuiint then 7 t I1tiHz tiltuii ti Theorem 1 A point 7 E QabMpq is a critical point for the action J 42gt its coordinate functions satisfy the Euler Lagrange equations BL d BL w a 0 16 Proof We prove only the implication and leave the other half which is quite a bit easier as an exercise We make the assumption that 7a 12 Q U where U is the domain of local coordinates as described above 16 For 1 S i S n let ab A R be a smooth function such that ia 0 ib and de ne a variation 1 766 gtlt ab A U by as t 11t s 1tulznt s ntl Let Wt ddt iti Then I Jds xit s it i i i x39it 31139Jiti i idt so it follows from the chain rule that b n 70 Z zilttgtiilttgtgtwilttgt zilttgtdcilttgtgt ilttgtl dt d E ltJltaltsgtgtgt Since ia 0 ib bndari bndari Lani 0a g lta gtdtamp g lta gt dtA gaiil dt andhence 1 M d M Z Thus if 7 is a critical point for J we must have 1 M d BL Z 0a 81i7lt8iigtl dtl for every choice of smooth functions In particular if 77 ab A R is a smooth function such that 77a0nb7 77gt0 on tub d g ltJltaltsgtgtgt and we set wt W zilttgtiilttgtgt 7 zilttgtiilttgtgtgtl 7 M d M 2 AW db Since the integrand is nonnegative it must vanish identically and hence 16 must hold then For a simple mechanical system the EulerLagrange equations yield a derivation of Newton7s second laW of motion lndeed i lt3 Z 9ijd1id1j7 ij1 17 then in the standard coordinates 117 i i 7 zn7 1391 i i 7 in l n v 705 i Z 9ij117mrnili7 7 15zl7iiizn i7j1 Hence a 7 1 n 69139 jk 19 81139 7E 81139 I I 781 k1 BL 7 n 7 d M agijvk w 2W k w 29 j1 j 1 11 where W d2zjdt2i Thus the EulerLagrange equations become n n n v 89 v 1 69 v 19 x39J 71146 7114677 29le l 81km I 7 2 V 81139 I I 81139 11 17k1 17k1 OI n 1 n 591 59 69k 615 7 J J 771 J 16 29111 2 jgl lt81 811 81139 I I 811 We multiply through by the matrix 917 Which is inverse to gij to obtain n v n 7 19 zl Z T kszk 7 Zgl 6117 17 jk1 11 Where n 99quot 6939 99k Fl h 7 Z 7 J i 1 8 Z 29 ark l 81 811 The expressions F j are called the Christo el symbolsi Note that if 117 i i 7 z are rectangular cartesian coordinates in Euclidean space7 the Christoffel symbols vanis i We can interpret the two sides of 17 as follows n il E F kijik accelerationl7 M71 n 8 7 2ng force per unit massli i1 811 Hence equation 17 is just the statement of Newton7s second laW7 force equals mass times acceleration7 for simple mechanical systems In the case Where 45 07 we obtain the differential equations for geodesics n Z rgkijik 07 1 9 jk1 where the F kls are the Christoffel symbolsi Example In the case of Euclidean space E with the standard Euclidean metric7 gij 61739 the Christoffel symbols vanish and the equations for geodesics become d2 139 I 0 dt2 The solutions are I I I 11 alt 1217 the straight lines parametrized with constant speed Note that the EulerLagrange equations can be written as follows d1 7 l W I 1 10 iii n l n 59 TEE Ejklrjk1z 2219 1312 This is a rstorder system in canonical form7 and hence it follows from the fundamental existence and uniqueness theorem from the theory of ordinary dif ferential equations 57 Chapter lV7 4 that given an element 1 E TpM7 there is a unique solution to this system7 de ned for t E is e for some 6 gt 07 which satis es the initial conditions 110 Ii 10 W0 ill In the special case where 45 07 we can restate this as Theorem 2 Given p E M and v E TpM there is a unique geodesic 7 z 676 A M for some 6 gt 0 such that 70 p and 70 1 Exercise II Consider the upper halfplane H2 E R2 z y gt 07 with Riemannian metric lt dz dz dy dy7 the so called Poincare upper half plane a Calculate the Christoffel symbols Pg bi Write down the equations for the geodesics7 obtaining two equations d2z d2y 7p 39 39 39 7 W cl Assume that y and eliminate t from these two equations by using the relatlon 2 7 3 Ed 7 4 dt2 dt dz dt dz dt dz dtQ Solve the resulting differential equation to determine the paths traced by the geodesics in the Poincare upper half plane 19 15 The LeviCivita connection In modern differential geometry the Christoffel symbols Pg are regarded as the components of a connection We now describe how that goes You may recall from Math 240A that a smooth vector eld on the manifold M is a smooth map XIMATM such that noXidM Where 7r TM A M is the usual projection or equivalently a smooth map X M A TM such that Xp E TpMi The restriction of a vector eld to the domain U of a smooth coordinate system I i i i I can be Written as XlUZfiai Where fizoen i1 If we evaluate at a given point p E U this specializes to X0Zfip 6 t A vector eld X can be regarded as a rstorder differential operator Thus if g M A R is a smooth function we can operate on g by X thereby obtaining a new smooth function Xg M A R by Xgp We let XM denote the space of all smooth vector elds on Mi It can be regarded as a real vector space or as an fMmodule Where fM is the space of all smooth realvalued functions on M Where the multiplication HM X MM A MM is de ned by MW fpXp De nition A connection on the tangent bundle TM is an operator v 2M gtlt 2M a 2M that satis es the following axioms Where we write VXY for VXY vfx yz fVXZgVyZ L11 vzfXgY ZfX szX Z9YszK 112 for fg e fM and XY Z e XMi Note that 419 is the usual Leibniz rule77 for differentiation We often call VXY the covariant derivative of Y in the direction of Xi Lemma 1 Any connection V is local that is ifU is an open subset of M XlUEO VxYlUEO and VyXlUEO 20 for anyY E Proof Let p be a point of U and choose a smooth function f M gt R such that f E 0 on a neighborhood of p and f E 1 outside Ur Then X U E 0 fX E Xi Hence VXYP VfXYP fPVXYP 07 VYXP VYfXP fPVYXP YfPXP 0 Since p was an arbitrary point of U we conclude that VXYNU E 0 and VyXlU E 0 This lemma implies that if U is an arbitrary open subset of M a connection V on TM will restrict to a unique wellde ned connection V on TUi Thus we can restrict to the domain U ofa local coordinate system 11 i i I and de ne the components Pg U A R of the connection y 8 n k 8 Vii311 Zrij 161 Then if X and Y are smooth vector elds on U say XZfii Yngji i1 8117 31 81 7 we can use the connection axioms and the components of the connection to calculate VXY n n V 89 n i V a VXY Z 1 an Z ijfjgk Wu 1 13 i1 j1 jk1 Lemma 2 VXY p depends only on Xp and on the values on along some curve tangent to X p Proof This follows immediately from 113 Because of the previous lemma we can VUX E T M whenever v E TpM and X is a vector eld de ned along some curve tangent to v at p by setting va when for any choice of extensions 7 of v and X of Xi In particular if 7 ab A M is a smooth curve we can de ne the vector eld VVM along 7 Recall that we de ne the Lie bracket of two vector elds X and Y by X 21 7 If X and Y are smooth vector elds on the domain U of local corrdinates 11 i i i I say Z a n v a XZf 291 i1 j1 then n iagji 1 91 9 XY 7121f 81 W 71219 W W Fundamental Theorem of Riemannian Geometry If M lt is a Rie mannian manifold there is a unique connection on TM such that l V is symmetric that is VXYi VyX XY for XY E XM 2 V is metric that is XltY Zgt ltVXY ZgtltY VXZgt forXY Z 6 This connection is called the Levi Civita connection of the Riemannian manifold M7 lt3 gt To prove the theorem we express the two conditions in terms of local coordinates 11 i i i I de ned on an open subset U of Mr In terms of the components of V de ned by the formula 6 n E 7 r197 114 ViiBx 61 1 1 81k the rst condition becomes 8 8 k k 39 l ij 7 l ji since 761 ifsz 7 0 Thus the l jls are symmetric in the lower pair of indices If we write lt Z gijdzi dzj ij1 then the second condition yields 8911 7 E E E 7 E E E 6 81k 7 81k lt81i7 azjgt 7 ltVaaxk 81139 7 azjgt l ltBzi 7V3axk azjgt n a a a n a n n 7 1397 7 7 l V7 V l I I l V ltgrklazl azjgt ltazi lgrmazzgt 129W ggllrkr In fact the second condition is equivalent to 89 n n all Zgljrlci 2911112 115 l1 l1 22 We can permute the indices in 115 obtaining a V n n g Zglkr j Zg r ki 1 16 l1 ll and n n a Zglir k nglr i 1 17 l1 ll Subtracting 115 from the sum of 116 and 117 and using the symmetry of F j in the lower indices yields agjk 69M agij n l 7 7 7 7 2 E F vi 811 l 81 81k l1 glk 1 Thus if we let gij denote the matrix inverse to gij we nd that 1 n 39 1991 991k 69 391 Fl 7 2 ll J J i1 2 i g lt61 61139 611 7 8 which is exactly the formula 18 we obtained before by means of Hamiltonls principle This proves uniqueness of the connection which is both symmetric and met rici For existence we de ne the connection locally by 114 where the F jls are de ned by 118 and check that the resulting connection is both symmetric and metrici Note that by uniqueness the locally de ned connections t together on overlaps In the special case where the Riemannian manifolds is Euclidean space lEN the LeviCivita connection is easy to describe In this case we have global rectangular cartesian coordinates 11 i i i IN on lEN and any vector eld Y on lEN can be written as N 6 37216 811 where f 3 HR 11 In this case the LeviCivita connection VE has components Pg 0 and there fore the operator VE satis es the formula 6 811 N VEY Zoo i1 It is easy to check that this connection which is symmetric and metric for the Euclidean metric If M is an imbedded submanifold of lEN with the induced metric then one can de ne a connection V XM gtlt XM A XM by VXYW V YpT7 23 where is the orthogonal projection into the tangent space Use Lemma 52 to justify this formula It is a straightforward exercise to show that V is symmetric and metric for the induced connection and hence V is the Levi CiVita connection for Mi Note that if 7 z a b A M is a smooth curve then VvWW WONT so a smooth curve in M g EN is a geodesic if and only if quy E 0 If we want to develop the subject independent of Nash s imbedding theorem we can make t e De nition If M lt is a Riemannian manifold a smooth path 7 ab A M is a geodesic if it satis es the equation quy E 0 where V is the LeVi CiVita connection In terms of local coordinates if n dzi o 7 E 7 7 l dt aw then a straightforward calculation yields n n V d2 11 o y dzJ o y dzk o y a 7 Flv if 7 119 V l a V M dt dt aw 11 k1 This reduces to the equation 19 we obtained before from Hamilton7s principlei Note that d 3W7 V WCW 2ltVM77 W 0 so geodesics automatically have constant speed More generally if 7 ab A M is a smooth curve we call quy the accel eration of 7 Thus if M lt is a simple mechanical system its equations of motion can be written as V WV gde 120 where in local coordinates gradq5 EgjinBzi asz 16 First variation of J Now that we have the notion of connection available it might be helpful to reView the argument that the function 1 b J QabMpq A R de ned by JW lt7tytgtdt has geodesics as its critical points and recast the argument in a form that is independent of choice of isometric imbedding 24 In fact the argument we gave before goes through with only one minor change namely given a variation 1 66 gt Q with corresponding 1 66 X a b A M we must make sense of the partial derivatives 8a 8a 3 7 8t 7 i i i 7 since we can no longer regard a as a vector valued function But these is a simple remedy We look at the rst partial derivatives as maps 8 8 673 766 gtlt ab A TM such that 7r 0 a7 7r 0 0A In terms of local coordinates these maps are de ned by n i ast i1 0451 We de ne higher order derivatives via the LeviCivita connectioni Thus for example in terms of local coordinates we set 8a n 82zk o a n k 8zi o a 8zi o a E V WS E I l 6th Elm O a as at 0 thereby obtaining a map Vaas 7ee gtlt ab A TM such that 7r ova35 a 8t 8t 8a 8a Vaaz 7 Vaaz 7 and so forth In short we replace higher order derivatives by covariant deriva tives using the LeviCivita connection for the Riemmannian metric The properties of the LeviCivita connection imply that a a Vaas Vaaz 25 Similarly we de ne 2V 616 61V 6 atas at aa as atas 33 a With these preparations out of the way7 we can now proceed as before and let it 676 A 9abMpyq be a smooth path with 10 7 and 8a 7 0 t V t 68 7 1 where V is an element of the tangent space T79 smooth maps V m b A TM such that 7r 0 Vt 7t for t 6 mb and Va 0 V02 ltStstdtgtl 50 bltv335 g 0t0tgt dt lt Then just as before7 d E ltJltaltsgtgtgt 5 S 7 a a viiat 0 t gm dt 5 lg ltZ ltotgtltotgtgt lt lt07tgtyvaxz9z ltovtgtgtl dt 8a 8a 0375 0 Ema d Since we obtain b ltJltaltsgtgtgt 50 7 ltvlttgt7 vw mw We call this the rst variation of J in the direction of V7 and write 1 MW 7 ltvlttgt7ltvmlttgtgtdtv 121 A critical point for J is a point 7 E QabMpq at which dJW 07 and the above argument shows that the critical points for J are exactly the geodesics for the Riemannian manifold M7 lt Of course7 we could modify the above derivation to determine the rst vari ation of the action I b Jltvgt wawww wow 26 for a simple mechanical system M gt We would nd after a short calcu lation that b b dJ7V ltvlttgt7ltvwgtlttgtgtdte d Vvtdt b 7 W WWgt6 i grad 7tgt th Once again the critical points would be solutions to Newton7s equation 120 17 Lorentz manifolds The notion of Riemmannian manifold has a generalization which is extremely useful in Einstein7s theory of general relatiVity as described for example is the standard texts 27 or 35 De nition Let M be a smooth manifold A pseudoRiemannian metric on M is a function which assigns to each p E M a nondegenerate symmetric bilinear map gtp TpM gtlt TpM H R which which varies smoothly with p 6 Mi As before varying smoothly with p E M means that if 45 11 i i I U A R is a smooth coordinate system on M then for p E U n 397 39gtp Z 9oltPgtdIllp 11le7 ij1 where the functions gij U A R are smooth The conditions that be symmetric and nondegenerate are expressed in terms of the matrix 91167 by saying that gij is a symmetric matrix and has nonzero determinanti It follows from linear algebra that for any choice of p E M local coordinates 11 i i i 1 can be chosen so that gtm 45 0 gt Iqu where Ipxp and Iqqu are p X p and q x 4 identity matrices with 10 4 n The pair p q is called the signature of the pseudoRiemannian metrici Note that a pseudoRiemannian metric of signature 0n is just a Rieman nian metrici A pseudoRiemannian metric of signature 1n 7 1 is called a Lorentz metrici A pseudoRiemannian manifold is a pair M where M is a smooth manifold and is a pseudoRiemannian metric on Mr Similarly a Lorentz manifold is a pair M where M is a smooth manifold and is a Lorentz metric on i Example Let Rnl be given coordinates t 11 i i i I with t being regarded as time and 11 i i i I being regarded as Euclidean coordinates in space and consider the Lorentz metric n lt gt 7chde 24H 3 M i1 where the constant c is regarded as the speed of light When endowed with this metric Rnl is called Minkawski spacetime and is denoted by an1i Four dimensional Minkowski spacetime is the arena for special relativity The arena for general relativity is a more general fourdimensional Lorentz man ifold M also called spacetimer In the case of general relativity the components gij of the metric are regarded as potentials for the gravitational forces In either case points of spacetime can be thought of as events that happen at a given place in space and at a given time The trajectory of a moving particle can be regarded as curve of events called its world line If p is an event in a Lorentz manifold M the tangent space TpM inherits a Lorentz inner product gtp TPM gtlt TPM R We say that an element v E TPM is 1 timelike if vvgt lt 0 2i spacelike if vvgt gt 0 and 3 lightlike if v vgt 0 A parametrized curve 7 ab A M into a Lorentz manifold M is said to be tirnelike if y is tirnelike for all u E a b If a parametrized curve 7 z a b A M represents the world line of a massive object it is tirnelike and the integral 1 b LW g 7ltwltuwltugtgtdu 122 is the elapsed time measured by a clock moving along the world line 7 We call Lgamma the proper time of 7 The Twin Paradox The fact that elapsed time is measured by the integral 122 has counterintuitive consequencesi Suppose that 7 z a gt L4 is a tirnelike curve in fourdimensional Minkowski spacetime parametrized so that 70 t711t712t713tl Then vii idii lt t wee d 7 ati1dtazi so 7 7 CH dt 2 28 Thus if a clock is at rest with respect to the coordinates that is dzidt E 0 it will measure the time interval 12 7 a while if it is in motion it will measure a somewhat shorter time interval This failure of clocks to synchronize is what is called the twin paradox Equation 123 states that in Minkowski spacetime straight lines maximize L among all timelike world lines from an event p to an event 4 When given an affine parametrization such curves have zero acceleration One might hope that in general relativity the world line of a massive body not subject to any forces other than gravity would also maximize L and if it was appropriately parametrized would have zero acceleration in terms of the Lorentz metric lt Just as in the Riemannian case it is easier to describe the critical point behavior of the closely related action 1 b szaimMzanR de nedby mj lt7 t77 tgtdt The critical points of J are called geodesics How does one determine the geodesics in a Lorentz manifold Fortunately the fundamental theorem of Riemannian geometry generalizes immediately to pseudoRiemannian metrics Fundamental Theorem of pseudoRiemannian Geometry If lt is a pesudo Riemannian metric on a smooth manifold M there is a unique connec tion on TM such that l V is symmetric that is VXYi VyX XY for XY E XM 2 Vis metric that is XltYZgt ltVXYZgtltY VXZgt forXY Z 6 The proof is identical to the proof we gave before Moreover just as before we can de ne the Christoffel symbols for local coordinates and they are given by exactly the same formula 118 Finally by the rst variation formula one shows that a smooth parametrized curve 7 ab A M is a geodesics if and only if it satis es the equation Vvfy E 0 As described in more detail in 35 there are two main components to gen eral relativity The Einstein eld equations describe how the distribution of matter in the universe determines a Lorentz metric on spacetime while time like geodesics are exactly the world lines of massive objects which are subjected to no forces other than gravity Lightlike geodesics are the trajectories of light rays 18 The RiemannChristoffel curvature tensor Let M lt be a Riemannian manifold or more generally a pseudoRiemannian manifold with LeviCivita connection Vi lf XM denotes the space of smooth vector elds on M we de ne RXM gtlt XM gtlt XM A XM by RXYZ VvaZ 7 VYVXZ 7 meZ We call R the Riemann Christo el curvature tensor of M lt Proposition 1 The operator R is multilinear over functions that is RfXYZ RX fYZ RXYfZ fRXYZl Proof We prove only the equality RXYfZ fRX YZ leaving the others as easy exerc1ses RX7YfZ VXVYfZ VYVXfZ VlXYfZ VXYfZ fVYZ VYXfZfVXZ X7YlfZ fVXYZ 7 X YfZ 01 sz XfVyZ fvxvyz 7 YXltfgtZ 7 X fwyZ 7 waxz 7 nyVyZ X7YlfZ fVXYZ fVXVyZ7 VyVXZ 7VXyZ fRXYZl Since the connection V can be localized by Lemma 51 so can the curvature that is if U is an open subset of M RXYZlU depends only XlU YlU and ZlUl Thus the curvature tensor is determined in local coordinates by its component functions jok U 7gt R de ned by the equations a a a n l a R w Proposition 2 The components R jk of the Riemann Christoffel curvature tensor are determined from the Christoffel symbols T k by the equations a a n n l l l l l Rijk aw iaxj PM 2 PM 2 PM iii m1 m1 The proof is a straightforward computation The simplest example of course is Euclidean space ENl In this case the metric coefficients gij are constant and hence it follows from 118 that the Christoffel 30 symbols Ti 0 Thus it follows from Proposition 82 that the curvature tensor R is identically zero Recall that in this case the LeviCivita connection VE on lEN is given by the simple formula E N a N a vx gfazigtxltfgt It is often easy to calculate the curvature of submanifolds on lEN with the induced Riemannian metric by means of the socalled Gauss equation as we now explain Thus suppose that L M 4 EN is an imbedding and agree to identify p E M with Lp 6 EN and v E TpM with its image L 1 E TplENi lf pEMandveTplEN we let v UT Ui where UT 6 TpM and ULLTpMi Thus is the orthogonal projection into the tangent space and is the orthogonal projection into the normal space the orthogonal complement to the tangent space We have already noted we can de ne the LeviCivita connection V XM gtlt XM A XM by the formula VXYP VigilPDT If we let XL denote the space of vector elds in lEN which are de ned at points of M and are perpendicular to M then we can de ne a XM gtlt XM A XLM by aX Y V Ypii We call a the second fundamental form of M in ENi Proposition 3 The second fundamental form satis es the identities afXY aXfY faX Y aXY aYXi lndeed altfX7Y Wer fV Yf faX7Y7 1X7fYVEMJ YDL XfY fVEYV faX7Y7 so a is bilinear over functions It therefore suf ces to establish aXY aY X in the case where XY 0 but in this case aXY 7 aYX VEY 7 VilaXi 0 There is some special terminology that is used in the case where 7 ab A M C E is a unit speed curve In this case we say that the acceleration WHO 6 T leN is the curvature of 7 while y tT VaM t geodesic curvature of 7 at t 31 7 L a7 t7 t normal curvature of 7 at it Thus if z E TpM is a unit length vector 11 1 can be interpreted as the normal curvature of some curve tangent to I at p Gauss Theorem The curvature tensor R of a submanifold M Q lEN is given by the Gauss equation ltRXYW Z 7 aX Z aY W 7 aX W aY Z 124 where X Y Z and W are elements of XM and the dot on the right denotes the Euclidean metric in the ambient space ENl Proof Since Euclidean space has zero curvature VEVEW 7 vv W 7 vamW 7 0 and hence 0 7 V VW z 7 VEVEW z 7 vamW z XVW Z 7 VgW V Z 7 YV W Z VEW VgZ 7 Van1W Z XltvyW Z 7 ltVyWVXZ 7 aY W aX Z 7 YWXWZ 7 ltVXWVYZgt 04X7W 39 0107 Z ltVXYVV7Zgt Thus we nd that 0 ltVXVYW Z 7 aY W aX W 7 ltVnyW 2 We W am W 7 Npmw 2 This yields VvaW7VyVXW7 VXyW Z aY W aX W 7aX W aY W which is exactly 124 For example we can consider the sphere of radius a about the origin in EMA Sna zlp l zn1 E En1 112 InT12 a2 If y 76 e 7 S Q En1 is a unit speed great circle say 7t acos1ate1 asin1ate2 where e1e2 are orthonormal vectors located at the origin in En1 then a direct calculation shows t at W 7 7 Nltvltm where Np is the outward pointing unit normal to S a at the point p E S a Thus the second fundamental form of S a in Enl satis es 11 1 ilN n for all unit length I E TpSnai a If I does not have unit length then I z 1 1 altH7H Wgt77Np WP ltIvIgtNp 7 By polarization we obtain 11 y 7711 ygtNp for all Ly E TpSnai Thus substitution into the Gauss equation yields 71 71 ltRltzygtw2gt lt7ltzzgtNltpgtgt lt7ltywgtNltpgtgt 7 iltz wgtNltgt ilt mm a 7 p a yv p A Thus we nally obtain a formula for the curvature of S a ltRltzygtwzgt ltltz72gtltwgt 7 ltz7wgtltyzgtgt In a similar fashion one can sometimes calculate the curvature of spacelike hypersurfaces in Minkowski spacetimer The metric coef cients for Minkowski space time an1 are constant so once again 1quot 0 and the curvature of Minkowski spacetime is zero In this case the LeviCivita connection VL is de ned by a N 6 a N Z a V foa LgfaIi Xf0aXfaIZA Suppose that M is an n dimensional manifold and L M A Lnl is an imbeddingi We say that LM is a spacelz39ke hypersurface if the standard Lorentz metric on anl induces a positivede nite Riemannian metric on Mi For sim plicity let us set the speed of light 5 1 so that the Lorentz metric on Ln1 is simply n lt gtL 7dt dt EM 8 dz i1 Just as in the case where the ambient space is Euclidean space we nd that the LeviCivita connection V XM gtlt XM A XM on TM is given by the formula VXYW V YpT7 33 where is the orthogonal projection into the tangent space If we let Xi denote the vector eld in lLN which are de ned at points of M and are perpen dicular to M we can de ne the second fundamental form of M in lL 1 by a 2M gtlt 2M A 257M by aX Y Milmy where is the orthogonal projection to the orthogonal complement to the tangent space Moreover the curvature of the spacelike hypersurface is given by the Gauss equation ltRX7YWv Zgt ltaX7 Z7aYv WM ltaX7 W 10 ZgtL7 125 where X Y Z and W are elements of As a key example we can set Hna tzlulzn E an1 z t2 7 112 7 7 In2 a2 t gt 0 the set of futurepointed unit timelike vectors situated at the origin in anll Clearly H a is an imbedded submanifold of lL 1 and we claim that the in duced metric on H a is positivede nitel To prove this we could consider 11 i i i I as global coordinates on H a so that t a2 I12 Then n I I d 21 zldzl a2 11 7mlt1ngt27 and the induced metric on H a is E zizjdzi dzj 1 1 n n 7Wdz dz dz dzl lt397 Thus I V E III a2 I12 7 1727 and from this expression we immediately see that the induced metric on H a is indeed positivede nitel Of course H a is nothing other than the upper sheet of a hyperboloid of two sheets Suppose that p E H a that e0 is a futurepointing unit length timelike vector such that p aeo and H is a twodimensional plane that passes through the origin and contains eol Using elementary linear algebra H must also contain a unit length spacelike vector e1 such that e0 e1gtL 0 Then the smooth curve gij 617 i 7 7e 6 7gt Hna de ned by 7t a coshtae0 asinhtae1 is spacelike and direct calculation shows that WW V tgtL sinh a coshta2 1 34 Moreover 1 1 7 t ltcoshlttagteo smutwet Nlttlttgtgt where Np is the unit normal to H a at 11 Thus WWWW WWW 07 so 7 is a geodesic and 1 QWOLWW 7L ENWO Note that we can construct a unit speed geodesic 7 in M as above with 70 p for any p E H a and 70 e1 for any unit length e1 6 TPH a Thus just as in the case of the sphere7 we can use the Gauss equation 1125 to determine the curvature of Hnai Thus the second fundamental form of H a in an1 satis es 11 1 iiN o for all unit length I E Tpllllnw7 where Np is the futurepointing unit normal to Mr If I does not have unit length7 then I z 1 1 a 77 7Np a 111 izter ltth W a l lt al l By polarization7 we obtain 11 y llt1ygtNp for all z y E TpNnai a Thus substitution into the Gauss equation yields ltRltzygtwzgt lt ltz72gtNltpgtgt lt ltytwgtN gtgt 7 wgtNltpgtgt lt ltyzgtNltpgtgtt Since Np is timelike and hence ltNpNpgtL 71 we nally obtain a for mula for the curvature of S a ltRltzygtwzgt ltltz72gtltwgt 7 ltz7wgtltyzgtgtt The Riemannian manifold H a is called hyperbolic space7 and its geome try is called hyperbolic geometry We have constructed a model for hyperbolic geometry7 the upper sheet of the hyperboloid of two sheets in an1 and have seen that the geodesics in this model are just the intersections with twoplanes passing through the origin in Lnli 19 Curvature symmetries sectional curvature The RiemannChristoffel curvature tensor is the basic local invariant of a pseudo Riemannian manifold If M has dimension n7 one would expect R to have n4 independent components R jk but the number of independent components is cut down considerably because of the curvature symmetries Proposition 1 The curvature tensor R of a pseudo Riemannian manifold M7 lt satis es the identities l RXY 7RYX 2 RX7 YZ RY7 ZX RZXY 0 3 ltRXYW7 Zgt 7ltRX YZ7 Wgt and 4 ltRX7 YW7 Zgt 7ltRVV ZX7 Ygt Remark 1 If we assumed the Nash imbedding theorem in the Riemannian case7 we could derive these identities immediately from the Gauss equation 124 Remark 2 We can write the above curvature symmetries in terms of the components R jk of the curvature tensor Actually7 it is easier to express these symmetries if we lower the index and write n l Rijlk ZglpRijk 121 This lowering of the index into the third position is consistent with regarding the Ri kls as the components of the map R TpM gtlt TpM gtlt TpM gtlt TpM A R by RXYZW ltRXYVVZgt In terms of these components7 the curvature symmetries are Rijlk 7 jilky Rijlk Rjkli Rkilj 07 Rijlk 7Rijkl7 Rijlk Rum Proof of proposition Note rst that since R is a tensor7 we can assume without loss of generality that all brackets of X7 Y7 Z and W are zero Then RX7 Y VXVY 7 VYVX VYvX 7 VXVY ROv XX establishing the rst identity Next7 RXYZ RY7 ZX RZXY vayZ 7 VyVXZ VszX 7 VszX Vzva 7 vazy VxVyZ 7 VzY VyltVzX 7 sz VzVXy 7 VyX 07 36 the last equality holding because V is symmetric For the third identity7 we calculate ltRX YZ zgt 7 ltvayz zgt 7 ltvyvxz Z 7 Xltvyz zgt 7 ltvyzvxzgt 7 Yltsz zgt 7 ltszvyzgt 7 xwz zgt 7 yxw zgt 7 XYltZ zgt 7 0 Hence the symmetric part of the bilinear form W Z H ltRXt YWt Zgt is zero7 from which the third identity follows Finally7 it follows from the rst and second identities that ltRX7YWt Zgt iltRYtXWt Zgt ltRXt WW7 Zgt ltRWt YXt Zgtt and from the third and second that ltRXYVV7 Zgt 7ltRXYZ Wgt ltRY7 ZX7 Wgt ltRZ7 XY7 Adding the last two expressions yields 2ltRX7 Y V7 Zgt ltRX7 WY7 Zgt RUV YX7 Zgt ltRY7 ZX7 Wgt ltRZ7 XY7 1 26 Exchanging the pair X7Y with W7 Z yields 2ltPltW ZX7Ygt ltRW XZt W ltRX7 ZWYgt ltRZYlV7 Xgt ltRY7 WZXgti 127 Each term on the right of 429 equals one of the terms on the right of 1 277 so ltRX7YWt Zgt ltRWZX7Ygtt nishing the proof of the proposition Proposition 2 Let RSTpMgtltTpMgtlt TpMXTpMHR be two quadrilinear functions which satisfy the curvature symmetries If Rz7 y z y 3z y z y for all z y E TpM then R 5 Proof Let T R 7 5 Then T satis es the curvature symmetries and TWWJW 07 for all Ly E TpM 37 Hence 0 TIyyzyz7yz TI7y717y TIyy7172T172717y TWJG 17 Z 2Tzyzz so Tzyzz 0 Similarly 0 Tz 2yz 210 Tzyzw Tz yzw 0 Tzwyzzw TzyzwTwyzzi Finally 0 2Tz y z w Tz y z w Tw y z z 2Tz yzw E Ty 21w E Tzyzw 3Tzyz SOT0andRSi This proposition shows that the curvature is completely determined by the sec tional curvatures de ned as follows De nition Suppose that a is a twodimensional subspace of TpM such that the restriction of lt to a is nondegeneratei Then the sectional curvature of a is R z 1 KW lt My gt 27 ltIyrgtltyyygtilt17ygt whenever zy is a basis for a The curvature symmetries imply that Ka is independent of the choice of basis Recall our key three examples the socalled spaces of constant curvature If M E then Ka E 0 for all twoplanes a g TpMi If M S a then Ka E la2for all twoplanes a Q TpMi If M H a then Ka E ElaQ for all twoplanes a Q TpMi The spaces of constant curvature are the most symmetric Riemannian manifolds possiblei De nition If M lt is a pseudoRiemannian manifold a diffeomorphism 45 M A M is said to be an isometry if lt pv ltvwgt for all 1 w E TpM and all p 6 Mr 128 Of course we can rewrite 128 as lt gt where ltv7wgt lt pv7 pwgt7 for Wu 6 TpMA Note that the orthogonal group On 1 acts as a group of isometries on S ai In this case we have an isometry group of dimension 12nn 1 Similarly the group of isometries of E the group of Euclidean motions is a Lie group of dimension 12nn I The group of isometries on H a also has dimension 12nn 1 it is called the Lorentz group In each of the three cases there is an isometry 45 which takes any point p of M to any other point 4 and any orthonormal basis of TpM to any orthonormal basis of Tin This allows us to construct nonEuclidean geometries for S a and H a which are quite similar to Euclidean geometry In the case of H a all the postulates of Euclidean geometry are satis ed except for the parallel postulatei 110 Gaussian curvature of surfaces We now make contact with the theory of surfaces in E3 as described in un dergraduate texts such as 29 If M lt is a twodimensional Riemannian manifold then there is only one twoplane at each point p namely TpMi In this case we can de ne a smooth function K M A R by Kp KTpM sectional curvature of TpMi The function K is called the Gaussian curvature of M An important special case is that o a twodimensional smooth surface M2 imbedded in R3 with M2 given the induced Riemannian metric We assume that it is possible to choose a smooth unit normal N to M N M2 A S2 with Np perpendicular to TpMi Such a choice of unit normal determines an orientation of M If NpM is the orthogonal complement to TPM then the second fundamental form a TpM gtlt TpM A NpM determined a symmetric bilinear form h TpM gtlt TpM A R by the formula May a17y39NP for any 6 Tlva which is also called the second fundamental form in the theory of surfaces Recall that if 1112 is a smooth coordinate system on M we can de ne the components of the induced Riemannian metric on M2 by the formulae 6 8 Qijlt7gtz forzvjilvg If F M2 A E3 is the imbedding than the components of the induced Rieman nian metric also called the first fundamental form are given by the formula 7 E g 7 81139 I 81 39 Similarly7 we can de ne the components of the second fundamental form by hij h 7 for i7j 1721 These components can be found by the explicit formula hij WE377 N Let a a Y Then it follows from the de nition of Gaussian curvature and the Gauss equation X that RX7YY7Xgt X7XgtltY7Ygt 7 X7Ygt2 7 aX7 X aY7Y 7 aX7Y aX7Y 7 ltX7XgtltY7Ygt ltX7Ygt2 h11 h12 7 h11122 7 h 2 7 h21 h22 911922 92 911 912 921 922 Example Let us consider the catemn39d7 the submanifold of R3 de ned by the equation 7 V12 y2 cosh27 Where 7 7 97 2 are cylindrical coordinates This is obtained by rotating catenary around the z axis As parametrization we can take M2 R X 5391 the and coshucosv FZRXSJHS by Xu7v coshusinv i u Here 1 is the coordinate on 5391 Which is just the quotient group RZ 7 Where Z is the cyclic group generated by 27L Then 6F sinhucosv 6F icoshusinv 67 sinh u sin 1 and 67 cosh u cos 1 7 u 1 v 0 and hence the coefficients of the rst fundamental form in this case are 911 l sinh2 u cosh2 u7 912 07 and 922 cosh2 ui The induced Riemannian metric or rst fundamental form in this case is lt cosh2 udu du d1 8 dv To nd a unit normal7 we rst calculate i sinh u cos 1 7 cosh u sin 1 7 cosh u cos 1 j sinh u sin 1 cosh u cos 1 7 cosh u sin 1 cosh u sinh u x Bu 8117 Thus a unit normal to S can be given by the formula Q X g 1 COS N 73 g 7 7 sinv 7 X 7 coshu Bu 3v s1nhu To calculate the second fundamental form7 we need the second order partial derivatives 82F 7 coshlu cos 1 82F 7 7 sir h u sin 1 BuQ 7 cos u sm 1 away 7 sm u cos 1 and 62x 7 cosh u cos 1 7 7 cosh u sin 1 8112 These give the coefficients of the second fundamental form 82F 82F h11w39N17 hi2h21m39N07 and WP hggimNiqi 71 K 7 cosh u4 Exercise III Consider the torus M2 6391 X 31 with imbedding 2 cos u cos 1 FIU7gtS by xuv 2cosusinv sin u where u and v are the angular coordinates on the two 5391 factors7 with u27r u7 v 27f vi a Calculate the components 917 of the induced Riemannian metric on M b Calculate a continuously varying unit normal N and the components hij of the second fundamental form of M c Determine the Gaussian curvature Ki 41 111 Review of Lie groups In addition to the spaces of constant curvature there is another class of man ifolds for which the geodesics and curvature can be computed relatively easily the compact Lie groups with biinvariant Riemannian metricsi Before discussing this class of examples we provide a brief review of Lie groups and Lie algebras following Chapters 3 and 4 of Suppose now that G is a Lie group and a 6 Ci We can then de ne the left translation by 0 LU CH C by LAT or a map which is clearly a diffeomorphismi Similarly we can de ne right transla tion RU CH C by RAT 70 A vector eld X on G is said to be left invariant if LUX X for all a E G where LaXf Xltf 0 Lu 0 L3 A straightforward calculation shows that if X and Y are left invariant vector elds on G then so is their bracket XYi See Theorem 79 in Chapter 4 of Thus the space gXE XG LUX Xfor alla 6G is closed under Lie bracket and the real bilinear map 1 l I g X g A g is skewsymmetric that is XY 7YX and satis es the Jacobi identity X K le Y7 lzlel Z X Yll 0 Thus g is a Lie algebra and we call it the Lie algebra of 0 If e is the identity of the Lie group restriction to TEG yields an isomorphism a g gt TeGi The inverse TEG gt g is de ned by a LUvi The most important examples of Lie groups are the general linear group GLnR n X n matrices A with real entries detA y 0 and its subgroupsi For 1 S ij S n we can de ne coordinates GLnlR A R by a Of course these are just the rectangular cartesian coordinates on an ambient Euclidean space in which GLnR sits as an open subset If X E GLnR left translation by X is a linear map so is its own differential Thus n i a n i k 9 LX afar W 8111 11 J 1 1 J 42 If we allow X to vary over GLn R we obtain a left invariant vector eld n a XA Z aleg ijk1 Na which is de ned on GLnRi It is the unique left invariant vector eld on GLnR which satis es the condition n i a XAIZajaz 7 zj1 J I 7 where I is the identity matrix the identity of the Lie group GLnRi Every left invariant vector eld on GLnR is obtained in this way for some choice of n X n matrix A a2 A direct calculation yields XAXB XAB where A B AB 7 EA 129 which gives an alternate proof that left invariant vector elds are closed under Lie brackets in this case Thus the Lie algebra of GLnR is isomorphic to glnR E TIC n X n matrices A with real entries with the usual bracket of matrices as Lie bracket Exercise IV Prove equation 129 For a general Lie group G if X E g the integral curve 9X for X such that 9X 0 6 satis es the identity 9X 3t 9X 8 9X t because the derivatives at t 0 for xed 8 are the same From this fact one easily concludes that 9X extends to a Lie group homomorphism 6X2RgtGi We call 9X the oneparameter group which corresponds to X 6 gr Since the vector eld X is left invariant the curve tH La9xt UOXO Rexz0 is the integral curve for X which passes through a at t 0 and therefore the oneparameter group of diffeomorphisms on G corresponding to X E g is 15 Rex for t E El In the case where G GLnR the oneparameter groups are easy to describe In this case if A E glnlR we claim that the corresponding one parameter group is 1 1 9m 6 ItA am Emu 43 Indeed it follows from the identity d LA LA LA dt 6 7 A6 7 e A that 9A is an integral curve for the left invariant vector eld determined by A and that 6A0 Al If G is a Lie subgroup of CLnR then its left invariant vector elds are de ned by taking elements of TIC Q TIGLnlR and spreading them out over G by left translations of Cl Thus the left invariant vector elds on G are just the restrictions of the elements of gln R Which are tangent to Cl We can use the oneparameter groups to determine Which elements ofgln R are tangent to G at L Consider for example the orthogonal group 0n A E GLrLlR ATA I Where denotes transposei lts Lie algebra is 0n A E glnR 61A 6 0n for all t E R Differentiating the equation ELATELA I yields etATATetA ELATAELA 07 and evaluating at t 0 yields a formula for the Lie algebra of the orthogonal group 0n A E glnR AT A 0 the Lie algebra of skewsymmetric matricesi T e complex general linear group GLn C n X n matrices A With complex entries det A y 0 is also frequently encountered and its Lie algebra is glnC E T80 n X n matrices A With complex entries With the usual bracket of matrices as Lie bracketi It can be regarded as a Lie subgroup of GL2nlRi The unitary group is Um A e sumo ATA I and its Lie algebra is un A E glnC AT A 0 the Lie algebra of skewHermitian matricesi With these basic ideas it should be easy to calculate the Lie algebras of most other commonly encountered Lie groups If G and H are Lie groups and h G A H is a Lie group homomorphism we can de ne a map h 19 A by hAX lheXel One can check that this is a Lie algebra homomorphism see Corollary 710 in Chapter 4 of This gives rise to a covariant functor77 from the category of Lie groups and Lie group homomorphisms to the category of Lie algebras and Lie algebra homomorphisms A somewhat deeper theorem shows that for any Lie algebra g there is a unique simply connected Lie group G with Lie algebra g This correspondence between Lie groups and Lie algebras often reduces problems regarding Lie groups to Lie algebras which are much simpler objects that can be studied via techniques of linear algebra 112 Lie groups with biinvariant metrics De nition Suppose that G is a Lie group A pseudoRiemannian metric on G is biinvariant if the difEeomorphisms L0 and R0 are isometries for every 0 6 Ci Example 1 For can de ne a Riemannian metric on GLnR by n ltgt 2 dz MI 130 ij1 This is just the Euclidean metric that GLnR inherits as an open subset of E The metric on GLnR is not biinvariant but we claim that the metric it induces on the subgroup 0n is biinvarianti To prove this it suf ces to show that the metric 130 is invariant under LA and RA when A E If A a E 0n and B E GLnlR then as o LAgtltBgt z31ltABgt ZzzltAgtz ltBgt Zazz B 161 161 so that n oLA 161 It follows that n L2dz Z agdzf 161 and hence Lid7 gt Z L2dz1 L2dz zj1 n n 39 k 39 l 39 39 k l E alcde aidzj E azag xj dzji ijkl1 ijkl1 Since ATA I ELI aZaf 6H and hence L2ltgt E 6kldz dzg lt gt jkl1 By a quite similar computation one shows that Rid7 gt lt3 gt7 for A 6 001 Hence the Riemannian metric de ned by 130 is indeed invariant under right and left translations by elements of the compact group Thus 130 in duces a biinvariant Riemannian metric on 0n as claimed Note that if we identify TIOn with the Lie algebra 0n of skewsymmetric matrices this Rie mannian metric is given y ltXYgt TraceXTY for XY e um Example 2 The unitary group Un is an imbedded subgroup of GL2nR which lies inside 0n and hence if lt is the Euclidean metric induced on GL2n R Lid gt12 lt3 gtE RM gtE for A e Um Thus the Euclidean metric on GL2n R induces a biinvariant Riemannian met ric on If we identify TIUn with the Lie algebra un of skewHermitian matrices one can check that this Riemannian metric is given by X Ygt 2Re ltTraceXTl7 for X Y E 1 31 Remark Once we have integration at our disposal we will be able to prove that any compact Lie group has a biinvariant Riemannian metric See 23 Proposition 1 Suppose that G is a Lie group With a biinvariant pseudo Riemannian metric lt Then H geodesics passing through the identity 6 E G are just the oneparameter subgroups of G E the LeVi CiVita connection on TC is de ned by VXY XY for XY e g P3 the curvature tensor is given by ltRXYW Z iltXY 2 Wgt for XY Z W e g 132 Before proving this we need to some facts about the Lie bracket that are proven in Recall that if X is a vector eld on a smooth manifold M with one parameter group of local diffeomorphisms 415 z t E R and Y is a second smooth vector eld on M then the Lie bracket XY is determined by the formula XiMp 7 g rYp 70 133 A De nition A vector eld X on a pseudoRiemannian manifold M lt is said to be Killing if its oneparameter group of local diffeomorphisms 475 Z t E R consists of isometries See the discussion surrounding Theorem 78 in Chapter 4 of 5 The formula 133 for the Lie bracket has the following consequence needed in the proof of the theorem Lemma 2 IfX is 3 Killing eld then ltVyX Z ltYVZXgt 0 forY Z 6 Proof Note rst that if X is xed ltVYX7ZgtCDgt and ltX7VYZgtP depend only on Xp and Ypl Thus we can assume without loss of general ity that Zgt is constant Then since X is Killing lt tY is constant an 72gt lt14 ltlt zgtvltzgtgt Hgt 7ltlX7Yl7Zgt 7 ltY Xv o aw t0 On the other hand since V is the LeviCivita connection 0 XltY Zgt VXYZgt ltYVXZgt Adding the last two equations yields the statement of the lemma Application If X is a Killing eld on the pseudoRiemannian manifold M lt and 7 ab A M is a geodesic then since ltVyX Ygt 0 d EWJQ ltVWCXgt WCVWQ 0 Thus lt7 Xgt is constant along the geodesic This often gives very useful con straints on geodesic We now turn to the proof of Theorem 1 First note that since the metric lt is left invariant XY6g ltXYgt is constant 47 Since the metric is right invariant each Rex is an isometry and hence X is a Killing eldi Thus ltVyXZgt VZXYgt 0 for XY Z 6 g In particular ltVXXYgt 4va X gmmo or Thus VXX 0 for X E g and the integral curves of X must be geodesics Next note that 0 VxyX Y VXX VXY VyX VyY VXY VyXi Averaging the equations VXYVyX0 ny7VyXXY yields the second assertion of the proposition Finally if X Y Z 6 g use of the Jacobi identity yields RXYZ VXVyZ 7 VyVXZ 7 VXyZ 1 1 1 1 1pc Y7Z11 7 1w X7 2 7 given12 jHani On the other hand if XY Z 6 g 0 2XltY7 Zl 2ltVXY7Zgt 2ltY7 VXZgt ltlX7Y17Zgt 07 X le Thus we conclude that 1 4 nishing the proof of the third assertioni 7 ltllX7Yl7Wl7Zgt iltlX7Yl7lZ7Wlgtv Remark If G is a Lie group With a biinvariant pseudoRiemannian metric the map 1 G A G de ned by 10 0 1 is an isometryi Indeed it is immediate that Ve 7id is an isometry and the identity 1 R071 0 1 0 L071 shows that 11 is an isometry for each a 6 G1 Thus 1 is an isometry of G Which reverses geodesics through the identity er More generally the map 0 L071 0 1 o 0 is an isometry Which reverses geodesics throug a A Riemannian symmetric space is a Riemannian manifold M lt such that for each p E M there is an isometry Ip M A M Which reverses geodesics through pi Examples include not just the Lie groups With biinvariant Rieman nian metrics and the spaces of constant curvature but many other important examples including the Grassmann manifolds to be described in the next sec tion 113 Grassmann manifolds If G is a compact Lie group with biinvariant Riemannian metric lt gt certain submanifolds M g G inherit Riemannian metrics for which geodesics and cur vature can be easily computed These include the complex projective space with its Fubini Study metric a space which occurs in algebraic geometry and other contexts To explain these examples we assume as known the basic theory of homo geneous spacesl As described in Chapter 4 9 of 5 if G is a Lie group and H is a compact subgroup the homogeneous space of left cosets GH is a smooth manifold and the projection 7r G A GH is a smooth submersionl Moreover the map G X GH A GH de ned by 0739H A 07H is smooth Suppose now that H is a compact subgroup of G and that s G A G is a group homomorphism such that 1 82 id and 2 Ho Gsaol Given such a triple G H s the group homomorphism s induces a Lie algebra homomorphism s g A g such that 33 idi We let hX6918XX7 pXegzsltXgt7X Moreover g 669p is a direct sum decomposition and the fact that s is a Lie algebra homomorphism implies that M Q h hm Q n M Q by Finally note that h is the Lie algebra of H and hence is isomorphic to the tangent space to H at the identity e while p is the tangent space to GK at EKUnder these conditions we can de ne a map f GH A G de ned by faH 03071 Indeed if h E H then fah ahsh 1a l 030 1 so f is a wellde ned map on the homogeneous space GHl Moreover 08071 73771 ltgt T la 37710 42gt 7 10 6 H so f is injectivel Finally one checks that X 6p tgt gtsequotX is a oneparameter group and checking the derivative at t 0 shows that 35quotX e X and hence fe X eQ Xl Moreover foe X oe XsequotXso 1 L0 0 R071e2 Xl 1 34 49 From these facts it follows that f is a onetoone immersion from GH into C and hence an imbedding which exhibits GH as an imbedded submanifold of G We can therefore de ne an induced Riemannian metric lt gtGH gtG such that ltX7YgtGH4ltX7YgtG7 for XY6pi 135 Since L0 0 R071 is an isometry7 the geodesics in the induced submanifold metric on GH are just the curves t gt gt fae Xi It follows that G acts as a group of isometries on GH when GH is given the induced metrici We mention two examples Example 1 Suppose G 0n and s is conjugation with the element pg lt7167gtltp Iqogtltqgt 7 where pq TL Thus 3A 11731411173 for A E 0n7 and it is easily veri ed that s preserves the biinvariant metric and is a group homomorphismi In this case H 0p gtlt 04 and the quotient OnOp gtlt 04 is the Gmssmann manifold of real p planes in n spacei Example 2 Suppose G Un and s is conjugation with the element if 0 I pxp gt where pqni I lt 0 Iqu In this case H Up gtlt Uq and the quotient UnUp gtlt Uq is the Grass mzmn manifold of complex p planes in n spacei The special case UnUl gtlt Unil of complex onedimensional subspaces of Un is also known as complex projective space Cpn li Theorem Given a triple 07H73 satisfying the above conditions the curva ture of is given by the formula ltRXYWZgt 4XY7 ZWgt7 for XYZW E T8KGH g p Sketch of proof The curvature formula follows from the Gauss equation for a submanifold M of a Riemannian manifold N7 lt when M is given the in duced submanifold metrici To prove such an equation one follows the discussion already given in li87 except that we replace the ambient Euclidean space lEN with a general Riemannian manifold N 7 lt ThusiprMgNandvETpN7welet v UT Ui where UT 6 TpM and 11LLTpM7 T and being the orthogonal projection into the tangent space and normal space The LeviCivita connection VM on M is then de ned by the formula WEMp ltV Yltpgtgti where VN is the LeviCivita connection on Ni If we let XLM denote the vector elds in N which are de ned at points of M and are perpendicular to M then we can de ne the second fundamental form a 2M gtlt 2M A XHM by aX Y V Ypii As before it satis ed the identities afXY aXfY faXY aXY aYXi If 7 ab A M Q lEN is a unit speed curve we call V y the geodesic curvature of 7 in N while T VfYVM Vf y geodesic curvature of 7 in M N l WWWl Under these circumstances one can show that the curvature tensor RM of M Q EN is given by the Gauss equation ltRMX7 WW7 Zgt ltRNX7 WW Zgtlt04ltX7 Z704Y7 Wgtlt0 X7 W7 0407 Z 1 36 17 7 normal curvature of 7 whenever X Y Z and W are elements of The proof of 136 is identical to the proof of the Gauss equation we gave before in In our application since geodesics in are geodesics in the ambient manifold G a 0 and the theorem follows directly from 136 together with 132 and the fact that the differential of the map f GH A G multiplies every element of p TeKGH by two Example We consider the special case in which G Un and s is conjugation with 71 0 I n 7 1 1 lt0 1ltn71gtxltn71gtgt so that the xed point set of the automorphism s is H U1 gtlt Un 7 l and GH CP L li Recall that the Lie algebra un divides into a direct sum un 669p where aXegzsltXgtX nXegrsXiX7 where h is the Lie algebra of U1 gtlt Un 7 1 We consider two elements 0 752 in 0 7m 7 520111 and Y WOIIO 5n 0 0 77 0 0 of p7 and determine their Lie bracket X7Y E It is easier to do this by carrying out the multiplication in matrix termsi To simplify notation7 we write 7 0 757 7 0 777T X 7 0 gt and Y7 lt77 0 7 137 7 751 77 0 7 TE 0 XY7lt 0 iE T 7 YXi 0 in 751 77 TE 0 X Y T i l l lt 0 E T n5 We next use the formula for the curvature of GH to show that the sectional curvatures Ka for CP 1 satisfy the inequalities a2 S Ka S 4a2 for some a gt 0 As inner product on TIUn7 we use so that and A7Bgt Re TraceATB 7 for A7B E This differs by a factor of four from the Riemannian metric induced by the natural imbedding into Ea 7 but with the rescaled metric ltX7 Yl RGQTW when X and Y are given by 137 To simplify the calculations7 assume that ltXXgt ltY7Ygt 17 and ltX7Ygt 0 Then 7 l5l2 lle 1 and 5T7 777 the latter since 5T7 is purely imaginary Then ltX7YL XviD 1 777T5 5 0 75T 77T5 0 Emcelt 0 ET W lt 0 EUT g T 2 lt4l1m quotll a EnTllifnT an 2 l1m5T l2 W2 Ichmf isl nl2sumltsgt The last expression ranges between 1 and 47 and it follows from the Cauchy Schwarz inequality that it achieves its maximum when 77 Thus if a is the twoplane spanned by X and Y7 4ltlX7lelX7Ylgt W Z 4 W 3 l1mlt5T gtl2l 52 lies in the interval 416 achieving both extreme values when n 7 1 2 2 The Riemannian metric we have de ned on GH CP 1 is called the Fubini Study metric It occurs frequently in algebraic geometry 114 The exponential map Our next goal is to develop a system of local coordinates centered at a given point p in a Riemannian manifold which are as Euclidean as possible Proposition 1 Suppose that M lt is a pseudo Riemannian manifold and p 6 M Then there is an open neighborhood V of0 in TpM such that ifv 6 TpM the unique geodesic 71 in M Which satis es the initial conditions 71 0 p and 71 0 v is de ned on the interval 01 Proof According to ODE theory applied to the secondorder system of differ ential equations dQIi dt2 n i dxjd kd 1621276 I t0 1 there is a neighborhood W of 0 in TpM and an e gt 0 such that the geodesic 7w is de ned on 06 for all w E Wt Let V eWi Then if v E V v em for some w E W and since 71 t Wu 6t 71 is de ned on 01 proving the proposition De nition De ne the exponential map exppV A M by exppv 7v Remark Note that if G 0n with the standard biinvariant metric lt which we constructed in li12 eprA em for A E TOni This explains the origin of the term exponential mapi77 Note that if v E V t gt gt expptv is a geodesic because expptv 711 71 t and hence expp takes straight line segments through the origin in TpM to geodesic segments through p in Mi Proposition 2 There is an open neighborhood U of 0 in TpM Which expp maps diffeomorphically onto an open neighborhood U ofp in M Proof By the inverse function theorem it will suf ce to show that exppbh I T0TpM TpM is an isomorphismi We identify To TPM with TpMi If v E TpM de ne Av R A TpM by Avt tvi 53 Then A 0 v and eXPp0v eXPp0L0 eXPp 0 MWO ltexppltwgtgt H gm v7 t0 so expp0 is indeed an isomorphismi It will sometimes be useful to have a stronger version of the above proposition proven by the same method but making use of the map exp neighborhood of 0section in TM a M X M de ned by expv p exppv for v E TpMi Proposition 3 Given a point pg 6 M there is an open neighborhood W of the zero vector 0 of TPOM Which eXp maps diffeomorphically onto an open neighborhood W of 100100 in M X M Proof If 0 denotes the zero vector in TPOM it suf ces to show that exp0 T0TM Tp0m0M X M is an isomorphismi Since both vector spaces have the same dimension it suffices to show that exp0 is an epimorphismi Let 7r12MgtltMgtM QIMXMHM denote the projections on the rst and second factors respectively Then 7r 0 exp TM A M is the bundle projection TM A M and hence 7r1 o exp0 is an epimorphismi On the other hand the composition TPOMQTMEMXMEM is just exp7O and hence 7r2 o exp0 is an epimorphism by the previous propo sitioni Hence exp0 is indeed an epimorphism as claimed Corollary 4 Suppose that M lt is a Riemannian manifold and pg 6 M Then there is an open neighborhood U ofpo and an e gt 0 such that epr maps 1 E TpM ltvvgt lt 62 diffeomorphically onto an open subset ofM for all p E U If is a Riemannian manifold and p 6 Mi If we choose a basis e1 i r en for TpM orthonormal with respect to the inner product we can de ne Euclidean coordinates 1391 i i z39 on TpM by 177 n z39ivai 42gt UZaieii i1 54 If U is an open neighborhood of p E M such that expp maps an open neighbor hood U of 0 E TpM diffeomorphically onto U we can de ne coordinates zllllznzUgtRn by xioexppz39il The coordinate 11 l l 1 are called Riemannian normal coordinates on M centered at p or simply normal coordinatesl These normal coordinates are the coordinates which are as Euclidean as possible near p Suppose that in terms of the normal coordinates ltgt Z gijdzi dzjl ij1 It is interesting to determine the Taylor series expansion of the 9177s in normal coordinates Of course we have gij p 61391quot To evaluate the rst order derivatives we note that whenever a constants the curve 7 de ned by 1 l l l a are Ii 0 7t at is a geodesic in M by de nition of the exponential mapl Thus the functions Ii Ii 0 7 must satisfy the geodesic equation n 55k 2 mini 0 ij1 Substitution into this equation yields n E Fgpala1 0i ij1 Since this holds for all choices of the constants a1 l l a we conclude that F jp 0 It then follows from 115 that 691739 81k P 0 Later we will see that the Taylor series for the Riemannian metric in normal coordinates centered at p is given y gij Elj 7 g E Rik o kzl higher order terms kl1 This formula gives a very explicit formula for how much the Riemannian metric differs from the Euclidean metric near a given point 11 Before proving this we will need the so called Gauss lemma 115 The Gauss Lemma We now suppose that 11 I are normal coordinates centered at a point p in a Riemannian manifold M7 lt and de ned on an open neighborhood U of p We can then de ne a radial function TUgtR by 7 xzl2zquot2 and a radial vector eld 5 on U 7 p by n Ii E S 7 For 1 S ij S n let Eij be the rotation vector eld on U de ned by a Na 139 I 77 BI 811 Eij Lemma 1 EijS 0 Proof This can be veri ed by direct calculation For a more conceptual ar gument7 one can note that the oneparameter group of local diffeomorphisms 45 t E R on U induced by Eij consists of rotations in terms of the normal coordinates7 so R so Em 7 g 450410 0 t0 Lemma 2 IfV is the Levi Civita connection on M then V55 0 Proof If a4 a are real numbers such that 2ai2 17 then the curve 7 de ned by I I 117t alt is an integral curve for 5 On the other hand7 W exp gait gt and hence 7 is a geodesic We conclude that all integral curves for S are geodesics and hence V55 0 Lemma 3 S S E 1 a 81139 Proof If 7 is as in the preceding lemma7 gm w 2ltvvmlttgtwgtgt o 56 so 7 ltt must have constant length But W03 W0 201152 17 i1 so we conclude that lt5 5 E 1 Lemma 4 lt5 E17 E 0 Proof We calculate the derivative of lt5 Eijgt in the radial direction 5lt57Eijgt ltV557Eijgt lt57V5Eijgt lt57V5Eijgt 1 lt5 VEW 5 EEijlt55gt 0 Thus lt5 Eijgt is constant along the geodesic rays emanating from 11 let ltXXgt Then as 11 1 gt lt00 llt57113ijgtlSllSllllEijllllEijllH 0 lf follows that the constant lt5 E17 must be zero Before proving the next lemma we observe that 5ltT 1 E170 0 These fact can be veri ed by direct computation Lemma 5 dr lt5gt in other words dTltX lt5 Xgt Whenever X is a smooth vector eld on U 7 Proof It clearly suf ces to prove this when either X 5 or X Eij In the rst case dTlt5 5ltT 1 lt55gt while in the second dTltEij E170 0 lt5Eijgt Remark It is Lemma 4 which is often called the Gauss Lemma 116 Curvature in normal coordinates Our next goal is to prove the following theorem which explains how the curva ture of a Riemannian manifold M lt measures deviation from the Euclidean metric Taylor Series Theorem The Taylor series for the Riemannian metric gij normal coordinates centered at a point p is given by 1 n gij tilj 7 E Z Rik o kzl higher order terms kl1 To prove this we make use of constant extensions77 of vectors in TpM relative to the normal coordinates 11 i i Suppose that w E TpM and 8 w Z a 7 i 1 all 17 Then the constant extension of w is the vector eld n 8 7 z W 7 a all i Since there is a genuine constant vector eld in TpM Which is expprelated to W W depends only on w not on the choice of normal coordinatesi We de ne a quadrilinear map GszMXTpMXTpMXTpM R as follows CIyyyzyw XYltZ7WgtP7 Where X Y Z and W and the constant extensions of z y 2 and w Thus the components of G Will be the second order derivatives of the metric tensori Lemma The quadralinear form C satis es the following symmetries l Cz y z w Cy z z w I 179727 w CIvvavz Cz z z I 0 Cz z z y 0 CIyywyw Czywyzyy and eeewm Czyyyzyw CIyzywyy CIywyyyz 039 Proof The second of these identities is immediate and the rst follows from equality of mixed partialsi The other identities require more wor i For the identity Cw www 0 we let W Eai8Bzi then the curve 7 de ned by I I I1 WW W 58 is an integral curve for W such that 70 pi It is also a constant speed geodesic and hence WWltWWgtp 0 We next check that Cw w 102 0 It clearly suf ces to prove this when 2 is unit length and perpendicular to a unit length wi We can choose our normal coordinates so that 8 8 We consider the curve 7 in M de ned by W 1107tt zio t 0 forigtli Along 7 we have W S and Z 111E12 so it follows from Lemma 124 that ltW Zgt E 0 along 7 and hence WWW zgtltpgt 0 it follows from the rst two symmetries that whenever u v E TpM and t E R 0 Gu tvu tvutvu7tv t3something t2Gv v u u 7 Cu u v 11 tsomethingi Since this identity must hold for all t the coef cient of it2 must be zero so Cu u v v Cv v u u which yields the fth symmetry To obtain the nal identity we let v1v2v3v4 e TpM and t1t2t3t4 e R and note that G Emtjvjyzmm 0 The coef cient of t1t2t3t4 must vanish and hence Z G v017v027 v06 va4 0A 0654 This together with the earlier symmetries yields the last symmetry 1 Now we let i p 811 i p 81 BIZ a gichl C y Lemma RiljkltPgt 9139ij 9139ij 7 Z7 Proof Since the Christoffel symbols 1quot vanish at p it follows that a z 710 w 81139 ijp 7 2 81139 81k 811 BIZ m7 9 l 7 1 9 99M 9ng 391k 811 7 2 811 81k hi 7 BIZ m7 and hence we conclude from Proposition 2 from 118 that l 52911 7 929m 7 929m 7 929139 2 aziazk aziazl azjazl azjazk p Rijlk 0 1 E QjLik gum ijil giljkl Qik giljk7 the last step following from the third symmetry of 01 From the last two lemmas7 we now conclude that RikjlltPgt RiljkltPgt 9139le 9139ij 9139ij 9139ij iggij b We therefore conclude that 82939 v 1 WW glRikMP Riljkpl Substitution into the Taylor expansion 1 n 82939 v gij SIj 5 16 mpzk1l higher order terms now yields the Taylor Series Theoremi 117 Riemannian manifolds as metric spaces We can use the normal coordinates constructed in the preVious sections to es tablish the following important result Local Minimization Theorem Suppose that Mn7 lt is a Riemannian manifold and that U is an open ball ofradius e gt 0 centered at 0 E TpM Which epr maps diffeomorphically onto an open neighborhood U ofp in M Suppose that v E U and that 7 01 gt M is the geodesic de ned by 7t eprtv Let L eprv If A 01 A M is any smooth curve With A0 p and M1 4 then H L 2 LW With equality holding only ifA is a reparametrization of 7 and E J 2 JW With equality holding only ifA 7 60 To prove the rst of these assertions we use normal coordinates 11111 I de ned on U Note that LW Tqi Suppose that A 01 A M is any smooth curve with A0 p and A1 qr Case I Suppose that A does not leave U1 Then 1 1 1 MA ltNlttgt Ntgtdt wow 2 wt wow 0 0 0 1 2 diew r o W 7 r 0 Mo Lm 0 Moreover equality holds only if A t is a nonnegative multiple of RAt which holds only if A is a reparametrization of 71 Case II Suppose that A leaves U at some time to 6 01 Then Lo ltNlttgtwlttgtgtdtgt HXlttgtHdt2 ltA lttgtRltAlttgtgtdt 2 0 WWW r a we 7 r o Agtlt0gt e gt Lm The second assertion is proven in a similar fashion If M lt is a Riemannian manifold we can de ne a distance function d M x M a R by setting dp q inf LW such that 7 01 A M is a smooth path with 70 p and 71 q A Then the previous theorem shos that dp q 0 implies that p qr Hence 1 dp q 2 0 with equality holding if and only if p q 2 d104 dqP7 and 3 dmi 3 ag dwi Thus M d is a metric space It is relatively straightforward to show that the metric topology on M agrees with the usual topology of M1 De nition If p and q are points in a Riemannian manifold M a minimal geodesic from p to q is a geodesic 7 ab A M such that 7a 107 71 4 and LW qul An open set U C M is said to be geodesically convex if whenever p and q are elements of U there is a unique minimal geodesic from p to q and moreover that minimal geodesic lies entirely within Ur Geodesic Convexity Theorem Suppose that M lt is a Riemannian manifold Then M has an open cover by geodesically convex open sets A proof could be constructed based upon the preceding arguments but we omit the details One proof is outlined in Problem 64 from 20 118 Completeness We return now to the variational problem with which we started this chapter Given two points p and q in a Riemannian manifold M does there exist a minimal geodesic from p to 4 For this variational problem to have a solution we need an hypothesis on the Riemannian metric De nition A pseudoRiemannian manifold M lt is said to be geodesically complete if geodesics in M can be extended inde nitely without running off the manifold Equivalently M lt is geodesically complete if expp is globally de ned for all p 6 Mi Examples The spaces of constant curvature E S a and H a are all geodesically complete as are the compact Lie groups with biinvariant metrics and the Grassmann manifolds On the other hand nonempty proper open subsets of any of these spaces are not geodesically completel Minimal Geodesic Theorem I Suppose that is connected and geodesically complete Then any two points p and q ofM can be connected by a minimal geodesic The idea behind the proof is extremely simple Given p E M the geodesic completeness assumption implies that expp is globally de ned Let a dp 4 then we should have 4 expp 121 where v is a unit length vector in TpM which points in the direction77 of qi More precisely let B5 be a closed ball of radius 6 centered at 0 in TpM and suppose that Be is contained in a an open set which is mapped diffeomorphically by expp onto an open neighborhood of p in Ml Let 35 be the boundary of B5 and let S be the image of 539 under exppl Since 5 is a compact subset of M there is a point m E S of minimal distance from q We can write m exppev for some unit length 1 E Tle Finally we de ne 7 0a A M by 7t expptvl Then 7 is a candidate for the minimal geodesic from p to q To nish the proof we need to show that 7a qr It will suf ce to show that MM a e t 138 62 for all t E 0ali Note that d7tq 2 a 7 t because if d7tq lt a 7 t then d104 S ew d7t74 lt H a it at Moreover if 138 holds for to E 0 a it also holds for all t E 0t0 because if t6 0t0 then d7t14 d7t77to d7t074 S to i t a t0 a i t We let to supt E 0 a dytq a 7 t and note that dyt0 q a 7 to by continuityi We will show that 1 to 2 e and 2 0 lt to lt a leads to a contradiction To establish the rst of these assertions we note that by the Theorem from dpq infdp39r d39rq 7 E S einfdrq 7 E S e dmq and hence a e dm q e d7e 97 To prove the second assertion we construct a sphere 5 about 7t0 as we did for p and let m be the point on S of minimal distance from qr Then d7to74 inf ldW oLT dT74 1T 6 5 6 dm747 and hence a7 to edmq so a7 t0e dmqi Note that dpm 2 to 6 because otherwise d104 S dp7mdm74 lt t06a t05a7 so the broken geodesic from p to 7t0 to m has length to e dpmi If the broken geodesic had a corner it could be shortened by rounding off the corner Hence m must lie on the image of 7 so 7t0 6 m contradicting the maximality of to It follows that to a d7a q 0 and 7a q nishing the proof of the theoremi For a Riemannian manifold we also have a notion of completeness in terms of metric spaces Fortunately the two notions of completeness coincide Hopf Rinow Theorem Suppose that M lt is a connected Riemannian manifold Then M d is complete as a metric space if and only if M lt 139s geodesically complete To prove this theorem suppose rst that M lt is complete as a metric space but not geodesically completei Then there is some unit speed geodesic 63 7 012 A M which extends to no interval 0b 6 for 6 gt 0 Let be a sequence from 012 such that t gt bi If pi 72 then dpipj g ti 7 tj so is a Cauchy sequence in M d Let p0 be the limit of Then by Corollary 4 from 1i14 we see that there is some xed 6 gt 0 such that exppl v is de ned for all v lt 6 when i is suf ciently large This implies 7 can be extended a distance 6 beyond p when i is suf ciently large yielding a contradiction Thus we need only show that when M lt is geodesically complete M d is complete as a metric space Let p be a xed point in M and a Cauchy sequence in M d We need to show that converges to a point 4 6 Mi e can assume that dqiqj lt e for some 6 gt 0 and let K dp q1i Then dpqi g K e for all i and hence qi eprv where S K 6 It follows that has a convergent subsequence which converges to some point 1 E TpMi Then 4 exppv is a limit of the Cauchy sequence 41 and M d is indeed a complete metric space 119 Smooth closed geodesics If we are willing to 1 t to A t we can give an other proof of the Minimal Geodesic Theorem which is quite intuitive and illustrates techniques that are commonly used for calculus of variations prob lems Moreover this approach is easily modi ed to give a proof that a compact Riemannian manifold which is not simply connected must possess a nonconstant smooth closed geodesici Simplifying notation a little we let QMpq smooth maps 7 01 A M such that 70 p and 71 q and let 9Mmq 7 6 9MP741J7lt a Assuming that M is compact we can conclude that there is a 6 gt 0 such that any p and q in M with dpq lt 6 are connected by a unique minimal geodesic 7W 01 A M with L7pq dpqi Moreover if 6 gt 0 is suf ciently small the ball of radius 6 about any point is geodesically convex and 717 depends smoothly on p and q If y z a ae A M is a smooth path and 2 elt then J7 a L7 2aelt6i a as we see from 12 Choose N E N such that 1N lt e and if 7 E 9M 1097 let pi 7iN for 0 S i 3 Ni Then 7 is approximated by the map Ry 01 A M such that W for te 51 64 Thus Ry lies in the space of broken geodesics77 BGNMpq maps 7 01 A M such that 71 7 is a constant speed geodesic 7 and 9M p q is approximated by BGNM P74a 7 6 BGNM P74 I JW lt a Suppose that 7 is an element of BGNM p74 Then if 39 2 Pi 7 7 then M71470 3 1 Fa lt J2Elt 67 so 7 is completely determined by pmplwvaiwvaN where 100 10 PN qt Thus we have an injection N71 A jzBC1391MpqagtMgtltMgtltgtltM7 jltvgtltvltagtwltgtgtl We also have a map 7 QMpqa gt BGNMpqa de ned as follows If 7 E QMpq 7 let TM be the broken geodesic from 1 N 7 1 10100 t0p17 N to to PN7139Y T toq We can regard TM as the closest approximation to 7 in the space of broken geodesicsi Minimal Geodesic Theorem II Suppose that Mn7 lt is a compact con nected Riemannian manifold Then any two points p and q ofM can be con nected by a minimal geodesic To prove this let M infUW I 7 6 9Mp74A Choose a gt M so that QMpqa is nonempty7 and let 7 be a sequence in QMpqa such that ij A 11 Let 3139 do the corresponding broken geodesic from 1 N71 pp01toplj7 N t0t0 PN71j7 T toqy 65 and note that J3j S J3ji Since M is compact we can choose a subsequence such that pm con verges to some point pi E M for each if Hence a subsequence of converges to an element 3 E BGNMpqai Moreover J 3139 S limped 31 S limped 7139 M The curve 3 must be of constant speed because otherwise we could decrease J be reparametrizing Hence 3 must also minimize length L on BGN M p 4 Finally 3 cannot have any corners because if it did we could decrease length by rounding cornersi This follows from the rst variation formula for piecewise smooth curves given in 1i3i2i We conclude that 3 01 A M is a smooth geodesic with dpq that is 3 is a minimal geodesic from p to q nishing the proof of the theoremi Remark Note that BGNM 104 can be regarded as a nitedimensional manifold which approximates the in nitedimensional space 9M 104 This is a powerful idea which Marston Morse used in his critical point theory for geodesicsi See 25 for a thorough working out of this approach Although the preceding theorem is weaker than the one presented in the previous section the technique of proof can be extended to other contexts We say that two smooth curves 3151HM and 32S391gtM are freely homotopic if there is a continuous path P 01 X 5391 gt M such that I 0t 31t and T1t 32ti We say that M is simply connected if any smooth path 3 5391 A M is freely homotopic to a constant pathi Thus M is simply connected if and only if its fundamental group as de ned in 14 is zero As before we can approximate the space MapSlM of smooth maps 3 z 5391 A M by a nitedimensional space where 5391 is regarded as the interval 01 with the points 0 and 1 identi ed This time the nitedimensional space is the space of broken geodesics77 BGNSlM maps 3 01 A M such that 3 R1 is a constant speed geodesic and 30 31 Just as before when a is suf ciently small then MaprM 3 e MapltskMgt JW lt a is approximated by BGNSIMa 3 E BGNSIM J3 lt a 66 Moreover if p 3iN then 3 is completely determined by P171027MP17M7PN Thus we have an injection N j BGN51Ma a 1W lt7 771 We also have a map 7 MapSlMa A BGN51Ma de ned as follows If 3 E MapSlMa let T3 be the broken geodesic from 1 N71 70 to p1vltNgt to to PN7139YltTgt to PN71 Closed Geodesic Theorem Suppose that M lt is a compact connected Riemannian manifold Which is not simply connected Then there is a noncon stant smooth closed geodesic in M which minimizes length among all noncon stant smooth closed curves in M The proof is virtually identical to that for the Minimal Geodesic Theorem ll except for a minor change in notationi We note that since M is not simply connected the space 7 3 E MapSl M 3 is not freely homotopic to a constant is nonempty and we let M infJ3 3 E 7 Choose a gt M so that f 3 E f J3 lt a is nonempty and let 3 be a sequence in f such that J3j A of Let 3139 T3j the corresponding broken geodesic and from 1 N71 PNj 70 t0 Pu 7 N to t0 PN71j 7 iN gt to PNj 71 and note that J3j g J3ji Since M is compact we can choose a subsequence such that pm con verges to some point pi E M for each if Hence a subsequence of converges to an element 3 E BGNSlMai Moreover JWJ S limjeooJWj S limjeooJWj Ma The curve 3 must be of constant speed because otherwise we could decrease J be reparametrizing Hence 3 must also minimize length L on BGN 51 M Finally 3 cannot have any corners because if it did we could decrease length by rounding cornersi This follows again from the rst variation formula for piecewise smooth curves given in li3i2i We conclude that 3 z 5391 A M is a smooth geodesic which is not constant since it cannot even be freely homotopic to a constant Chapter 2 Differential forms 21 Tensor algebra The key advantage of differential forms over more general tensor elds is that they pull back under smooth maps In the next several sections we explain how this leads to one of the simplest ways of constructing a topological invariant of smooth manifolds namely the de Rham cohomologyi Recall that TM is the vector space of linear maps a TpM A R We de ne the kfold tensor product kT5M to be the vector space of Rmultilinear maps k qszpMXTpMX gtltTpMgtlRi Thus 111sz is just the space of linear functionals on TpM which is TM itself while by convention 0T1 M R We can de ne a product on it as follows If 45 E kTgM and 1 E ZTM we de ne lt15 1 E kszfM by v1iu vkl 45111 ivk vk1u ivkli This multiplication is called the tensor product and is bilinear lta a3gt w amwww a 113 amwabw as well as associative lt15 1J w q5 wi Hence we can write q5 1b 8 Ad with no danger of confusion The tensor product makes the direct sum 00 t s 7 1c Tp M 7 So TpM i0 into a graded algebra over R called the tensor algebra of Tszi 68 Proposition 1 If zlpiwzn are smooth coordinates de ned on an open neighborhood ofp E M then dzi1p dziklp1gi1 ltnm1lt ik gn is a basis for kTM Thus szfM has dimension nk Sketch of proof For linear independence suppose that Zairikdzil 1 dziklp 0 Then 8 pinaxj 17 F i a i a Zailikdz 1 1 p dz kl 17gt ajljki To show that the elements span suppose that 45 E kTM and show that 7 We let AszfM denote the space of skewsymrnetric elements of lt15 6 szfMi By skewsymmetric we mean that the value of 45 changes sign when two distinct arguments are interchanged 8 0 Zailikdzlllp dzlklp 45 Zailikdzi1lp dziklp 8 8 where 2 45 7 81 p 451117m7vi739 7vj7kagt v17wvj7quot39 aviauka7 whenever i y j This can be expressed in terms of the symmetric group 5k on k lettersi Recall that by de nition 5k is the group of bijections from the set 1 i i i 16 onto itself with composition being the group operation We de ne a function sgn 5k gt i1 by sgna 11 and check that it is a group homomorphismi We say that an element a E 3k is even if sgna 1 add if sgna 71 Then a multlinear map k 2TpMXTpMXgtltTPMgtR is skew symmetric if 4151100 i v0k sgna v1i ivk 69 for all a 6 Ski Note that AkT5M is a linear subspace of kTZ Mi We de ne a projection Alt szfM A AszfM by l Alt v17ka H E Sgn0va17vak 765k It is a straightforward exercise to show that A1tA1t 1 w A1tqb 1 w A1tqb A1t my 21 For details7 one can check the argument for Lemma 66 of Chapter 5 in If 45 E AszfM and 1 6 AngM we can de ne 45 A11 6 AkHTsz by kl M1 km Alt 15 This multiplication is called the wedge product It is bilinear7 a45lt15M1 awwwzp WWW wWWW skewcommutative am 1p 70 ab for ab 6 AkTgM and 1i 6 AngM and associative 45M1 Aw 4 WWW Only the last fact is nontrivial7 and it follows rather quickly from identity 21 This product makes the direct sum k A TpM EA Tp M i0 into a graded commutative algebra over R called the exterior algebra of Tszi Proposition 2 If manqzn are smooth coordinates de ned on an open neighborhood ofp E M then drillpAHAdziklpl i1lti2lt ltik n is a basis for kTM Thus kTM has dimension the proof is quite similar to that of Proposition 1 22 The exterior derivative We now let AkTM UAkTM a disjoint union Just as in the case of the tangent and cotangent bundles A T M has a smooth manifold structure together with a projection 7r AkTM A M such that 7rAkTM pi We can describe the coordinates for the smooth structure on AkTM as follows If 11 i i i 1 are smooth coordinates on an open set U E M the corresponding smooth coordinates on 7r 1U are are the pullbacks of 11 i i i I to 7r 1U together with the additional coordinates pilik 7r 1U A R de ned by j1ltquot39ltjk 391 39k 7 1011 E ajljkdz lpAnAdz 17gt fairniki If U is an open subset of M a di erential kform or a dz erential form of degree k on U is a smooth map w U A AkTgM such that 7r 0 w idUi lnformally we can say that a differential kform on U is a function w which assigns to each point p E U an element wp E AszfM in such a way that wp varies smoothly with 11 Let QWU denote the real vector space of differential kforms on Ur If U 11 i i i is a smooth coordinate system on M we can de ne dzi1 dz 6 GNU by dzi1 Adzikxp dziilp dziklpi Then any element w E 9k U can be written uniquely as a sum w Z filikdzi1 Adz i1lt39ltik where filmik U A R is a smooth function If w E 9k U and 45 E QZU then we can de ne the wedge product w 45 E QklU by W A 4910 10 A W Note that if f E 90M fM and w E QWM then f Aw fwi Exterior Derivative Theorem There is a unique collection of linear maps of real vector spaces d QWM A Qk1M Which satisfy the following conditions 1 Ifw is a k form the value dwp depends only on w and its derivatives at p 2 Iff is a smooth real valued function regarded as a differential 0 form is the differential of f de ned before 3 dod0 4 Ifw is a k form and 45 is an l form then dw A lt15 dw A lt15 1kw A L145 We call d the exten39or den39vatz39vei We begin the proof of the theorem by establishing uniquenessi By property 1 it suf ces to prove uniqueness in the case where where U is the domain of a local coordinate system 11 i i i lf w E 9k U we can write w E filikdzi1 AAdziquot i1lt39ltik where firmik U A R is a smooth function Using linearity and the axiom for products we now nd that dw Z dfi1ikdzi1AAdzik i1ltquot39ltik Z dfilikAdzi1AAdzik Z filikddzi1AmAdzik i1ltltik i1ltquot39ltik Using the axiom for products the fact that d o d 0 and induction one shows that ddzi1 Adz 0 Hence dw Z delk A dz A A dz 22 1391 lt 39ltik where dfirwk E 91U is the previously de ned differential of a function This formula establishes uniqueness We next prove local existence existence on U where U is the domain of a local coordinate system 11 i i 1 To do this we can de ne dw by 22 and check that it satis es the axioms The rst two axioms are immediate To establish the last axiom we use the easily proven formula dfy 9 fdy Suppose that w E filikdzi1AAdzik andqb Z gjijdzj1AAdzj i i1ltquot39ltik j1ltquot39ltjt Then w lt15 Zfil kgjwjtdril A I I I Adzik Adzjl A I I I Adxjty and hence do A Zdf1kgj1jAdz AAdzik A dsz A Admit 291jdf1ik Adaci1 A Adxiquot Adacj1 A Adzj Zfilikdgj1j Adaci1 A Adz Adacj1 A Adacjt dlqu57lkwAdq5i For the third axiom we use the equality of mixed partial derivatives First we note that if f E 90 M then n 5f 1 7 n 82f i j ddf 7 d 61de gt 7 1 8161de Adz M 82f 62f V 7 7 7 Z J Z 91er azjazi dI Adm 0 zltJ In general if w E QWU say w Z fiiikdzi1 A Adz i1ltquot39ltik we nd that ddw dlt Z d101 A dz A A dz i1lt39ltik Z ddfi1ikAdzi1A dzi 7 Z dfilikAddzi1A Adzik 0 i1ltquot39ltik 71ltquot39ltik This nishes the proof of local existence To prove global existence we note that the locally de ned exterior derivative operators must t together on overlaps due to uniqueness and hence they t together to form a globally de ned exterior derivative operator on M i Example The exterior derivative is actually an extension of the gradient divergence and curl operators one meets in several variable calculus Thus suppose that M E3 with the standard euclidean coordinates zyz and let dx dzi dyj dzk NdA dy dzi dz dzj dz dyk where ij k is the usual orthonormal basis If f 6 HOURS is a smooth function df dz 6f 9f BI aiydy Edz 7 gradlent of dxi lf 9 F dx is an element of SPURS where F is a vector eld7 say 9 sz Qdy Rdz then 6R 6Q d9 677 dyAdz 8P 8R 8Q 8P ltE7Egt dzAdzltEiagt dzAdy curl of F NdAi Finally7 if w F NdA is an element of SPURS say wdeAdzQdyAdzRdzAdy then do E dz dy d2 divergence of Fdz dy dzi 81 By 82 The exterior derivative extends these familiar operations from calculus to ar bitrary smooth manifolds in such a way that they are natural under smooth mapsi77 We now explain what we mean by natural under smooth mapsi Suppose that F M A N is a smooth mapi lfp 6 M7 the linear map F017 I TPM TFPN induces a linear map k k Fp iA TFpNgt A TpM By F v17vk Fpv17FpvkA This in turn induces a linear map F MN a WM by Fltwgtltpgt FltwltFltpgtgtgt lff E 90M7 we agree to let f oFi Proposition The map F preserves wedge products and exterior derivatives 1 Fw A9 Fw F9 2 dFw Fquot We leave the proof of the rst of these facts as an easy exercise We rst check the second for the case of a function f E 90 In this case7 if v E TpM7 Fdfv dfFpv Fpvf vf 0 F df 0 Fv dFfv 74 We next check in the case where N U the domain of a local coordinate system 11 i i i 1 In this case w Z fiiikdzi1 A Adzik i1lt39ltik and I I dw Z delk Adz A Adam i1lt39ltik Using the rst assertion of the proposition we see that Fdw Z Fdf1k A Fdzi1 A A Fdzi i1ltquot39ltik Z dfi1ik 0F dzi1 o F Adzik o F 23 i1lt39ltik On the other hand Fm Z fl1quot 0Fdri1 OF A Ada 0F i1ltquot ltik dFm Z df1k 0F Adzi1 0F A Adm 0F 24 i1lt39ltik Comparing 23 and 24 we see that Fquot o d d o F when N has a global coordinate systemi Since the operators are local Fquot o d d o Fquot on any smooth manifold Mi lf w E and X1 m Xk are smooth vector elds on M we can de ne a smooth function wX1i i Xk on M by wltX17Xkgtpgt WPX107Xk Exercise V Show that if X and Y are smooth vector elds on M and t9 6 91M is a smooth oneform then dt9093 XWY YWX 9lX7Yl Hint Use local coordinatesi 23 Integration of differential forms The way to think of differential forms of degree n is that they are integrands for multiple integrals over n dimensional oriented manifolds Two smooth charts U 11 i i and V ylp i on an n dimensional smooth manifold M are said to be coherently oriented if Byi det ltazjgt gt 0 75 where de ned We say that M is orientable if it possesses an atlas of coherently oriented chartsl Such an atlas is called an orientation for Ml An oriented smooth manifold is a smooth manifold together with a choice of orientation Suppose that M is a smooth manifold with orientation de ned by the atlas A of coherently oriented chartsl A chart V7 yl7l i i 7y on M is said to be positively oriented if Byi d t 7 0 e lt 61 gt 7 where de ned7 for every chart U7 117 i i i 7 in Al Local integration of n forms Suppose that w is a smooth n form with com pact support in an open subset U of a smooth n dimensional oriented manifold Ml If 45 117 i i i 7 I are positively oriented coordinates on U7 we can write fdzl Adz2 Adznl We can then de ne the integral of w over U by the formula Aw 045754717777 Thus to integrate an n form over an oriented n manifolds7 we essentially we just leave out the wedges and take the ordinary Riemann integrall We need to check that this de nition is independent of choice of positively oriented smooth coordinatesl To do this7 note that if 1 y1 n 39 7 i i i 7 y 1s a second positively oriented coordinate system on U7 then n 61139 dzl dyj7 and hence n n 811 81 v v 1 n 7 n dz Adz 7 267ndywnwy 111 Jn1 811 Br 7 01 0n gs gym ay0ndy y 811 Br 1 n sgnaWWdy Adyi Recall that ifA a is an arbitrary n X n matriX7 det A Z sgnaa7171 2007 765 and since the two coordinate systems are coherently oriented7 i dzlAdzndetltay I gtdylAnAdy ayi 1 n detltaxjgt dy Adyl 76

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