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# INTRO DIFF GEOMETRY MATH 240A

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Lectures on Differential Geometry Math 2400 John Douglas Moore Department of Mathematics University of California Santa Barbara7 CA7 USA 93106 e mail moore mathuosbedu May 97 2011 Preface This is a set of lecture notes for the course Math 240C given during the Spring of 2011 The notes will evolve as the course progresses Contents H N Riemannian geometry 1 111 Review of tangent and cotangent spaces i i i i i i i i i i i i i i 1 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 4 1 3 Geodesics i i i i i i i i i i i i i i 8 11311 Smooth paths i i i i i i i i 8 11312 Piecewise smooth paths i i i i i i i i i i i i i i i i i i i 12 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 13 115 The LeviCivita connection i i i i i i i i i i i i i i i i i i i i i 19 116 First variation of J intrinsic version i i i i i i i i i i i i i i i i 25 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 28 118 The RiemannChristoffel curvature tensor i i i i i i i i i i i i i 31 1 9 Curvature symInetries sectional curvature i i i i i i i i i i i i i 39 1 10 Gaussian curvature of surfaces 42 1111 Matrix Lie groups i i i i i i i i i 46 1 12 Lie groups With biinvariant metrics i i i i i i i i i i i i i i i i i 51 1 13 Projective spaces Grassmann manifolds i i i i i i i i i i i i i i 56 Normal coordinates 62 211 De nition of normal coordinates i i i i i i i i i i i i i i i i i i 62 212 The Gauss 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 66 2 3 Curvature in normal coordinates i i i i i i i i i i i i i i i i i i 68 2 4 Riemannian manifolds as metric spaces i i i i i i i i i i i i i i i 73 2 5 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 74 26 Smooth closed geodesics i i i i i i 77 27 Parallel transport along curves 81 2 8 Geodesics and curvature i i i i i i i i i i i i i i i i i i i i i i i 82 Bibliography 86 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 1 or in 7 Suppose that M is a smooth manifold and p E M and that fp denotes the space of pairs U f Where U is an open subset of M containing p and f z 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 functional 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 17 71 139 Once again under change of coordinates the basis elements transform by the chain rule 239 7 n ayi j dy lp gg df lp 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 erentz39al 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 a I dflp Z a mu il 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 A M de ned by 7rTpM pi lfqb 11 i i I is a coordinate system on U C M we can de ne a corresponding coordinate system 45 zliuznp1iupn on n 1U C TM by letting n n V a V 11 E a w 11P7 Pi E 039deij ai F1 17 j1 It is customary to use P1Hipn to denote momentum coordinates on the cotangent space For the various choices of charts U on M the corre sponding charts n 1U on TM form an atlas making T M into a smooth manifold of dimension Qni 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 A 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 Riemanm39zm 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 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 R and we say that lt gtp varies smoothly with p if the functions gij are smooth We call the functions gij the compo nents of the Riemannian metric with respect to the coordinate system 45 11 i i i 1 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 lt Z gijdzi dzji ij1 If 1 yl i i y is a second smooth coordinate system on V E M with lt3 39gtlV Z hijdyi dyj7 ij1 it follows from the chain rule that on U N V n Byk Byl gij Z Maggi kl1 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 formula gij hkl avltwgt ltv w In terms of components if v Zai all then av gijpajdzilpi i1 p 11 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 n I n V a if a Zaidzllp then 3900 2 WWW all 1 1 7 z 17 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 The simplest example of a Riemannian manifold is ndimensional Euclidean space E which is simply R together with its standard rectangular cartesian coordinate system 11 i i i 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 nianva nia 71677 lt2 61 Zb1pgtlt WP gag 712w 17 j1 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 by ltvwgtp pvFpw forvw6TpMi Here F 7 is the differential of F at p de ned in terms of a smooth coordinate system 45 11 i i i I by the explicit formula 6 F 81139 p Clearly ltvwgtp is symmetric and it is positive de nite because F is an immer sioni Moreover 8 8 Fquotlti gt39Fquotlti39 pgt 81 p 81 p 91 DiF o 1 p DjF o 1 p so gij p depends smoothly on p Thus the imbedding F induces a Riemannian metric lt on M and we write lt3 Flt397 gt DiF 0 1 p 6 RN 3 p7 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 isometn39c if lt Flt The local problem of nding an isometric imbedding is equivalent to solving the nonlinear system of partial differential equations 8F 8F 7 H 811 azj 79117 where the 9177s are known functions and the Envalued function F is unknown Unfortunately this system does not belong to one of the standard types ellip tice parabolic hyperbolic that are studied in PDE theoryi Nash s Imbedding Theorem If M lt is any smooth Riemannian mani fold there exists an isometric imbedding F M gt EN into Euclidean space of some dimension This was regarded as a landmark theorem when it rst appeared 11 The proof is dif cult involves subtle techniques from the theory of nonlinear partial differential equations and is beyond the scope of this course A special case of Example 2 consists of twodimensional smooth manifolds which are imbedded in Egi 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 Lecons sur la theorie generate des surfaces et les applications g om trz39ques du calcul in nitesimal by Jean Gaston Darbouxi 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 W 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 n dzi E 7 7 7 t H dt azi E T7EM for t E a 12 yt Then the length of 7 is given by the integral 1 Lo ltwlttwlttgtgtwti We can also write this in local coordinates as b n dzi dzj Lo 2 WW d War 1 ij1 Note that if F M gt 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 It is the foundation for the nonEuclidean geometry discovered by Bolyai and Lobachevsky in the nineteenth century 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 tab A M I WI lamb q We can de ne two functions L J QabMpq A R by b b Lv lt7 t77 tgtltzgtdtv Jltvgt ltv lttgtv lttgtgtltzgtdt Although our geometric 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 family 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 4 At rst we use the notion of in nitedimensional man ifold somewhat informally in more advanced courses on global analysis it is important to make the notion precise Proposition 1 L72 S 21 7 aJ7 Moreover equality holds if and only if MOM75 is constant if and only ify has constant speed Proof We use the CauchySchwarz inequality b 2 LW l WMamt S b b dtl l ltv tw tgtdtl 2baJvA 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 6 9pm M p 4 2b a WW S Ll2 S 21 7 aJ7 and hence Jy 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 LW Approximate A by an immersion A1 such that LA1 lt LW this is possible by a special case of an approximation theorem due to Whitney see 6 page 27 Since the derivative X1 is never zero the function 3t defined by slttgt wow is invertible and A1 can be reparametrized by arc length It follows that we can find an element of A2 a b A M which is a reparametrization of A1 of constant speed But then 2b aJ2 L2l2 L1l2 lt MW S 2b 7 WM 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 L We want to develop enough of the calculus on the infinite dimensional manifold77 QabMpq to enable us to find the critical points of J This is exactly the idea that Marston Morse used in his celebrated calculus of variations in the large77 a subject presented beautifully in Milnor7s classical book To start with7 we need the notion of a smooth curve 3 z 7676 A QabMpq such that 30 7 where 7 is a given element of 9 In fact7 we would like to de ne smooth charts on the path space QabM p q7 but for now a simpler approach will suf ce We will say that a variation of 7 is a map it 676 A Maudhag such that 10 7 and if a 676 X a7b A M is de ned by ast d8t then a is smooth De nition An element 7 E QabMpq is a critical point for J if di Jd 07 for every variation 1 of 7 1 3 3 50 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 generalization of the notion of straight line from Euclidean to Riemannian geometryi 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 ENi Thus we imagine that M Q EN and thus each tangent space TpM can be regarded as a linear subspace of RNi Moreover7 ltvwgtp vw7 for vw ETPM GEN where the dot on the right is the dot product in ENi If d1 575w 9abMpyq is a variation of an element 7 E QabM p7417 with corresponding map a z 7676 gtlt a z HMQEN then d E ltJltaltsgtgtgt d 1 7 6a 6a 50 7 E a 5a Estdt 50 I 2 E 1 8a a 7636 s t Estdt b 2 E 1 8a 70 a W 39 50th 10 where a is regarded as an lENvalued function If we integrate by parts and use the fact that a a a 1 E03717 0 Ema we nd that 1 2a b gweo 7 3w gmwwelvwwwmw an 50 where Vt Ba83 0t is called the van39atz39zm eld of the variation eld 6A Note that Vt can be an arbitrary smooth lENvalued function such that Va 0 Vb 0 Vt E Tqu for all t 6 ab that is V can be an arbitrary element of the tangent space77 T79 smooth maps V a b A lEN such that Va 0 Vb Vt E TymM for t 6 ab We can de ne a linear map dJW TVS A R by b d WWW ltVt77 tgtdt EUOWS 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 TymM for all t E abi In other words 7 ab gt M is a geodesic if and only if 7T 0 for all t e w 15 where y tT denotes the orthogonal projection of 7 t into Tqui To see this rigorously we choose a smooth function 77 z a b A R such that na0nb7 77gt0 on tub and set T V0 7W WNW 7 which is clearly an element of T751 Then dJW V 0 implies that b T b T 2 nmwm wmaanwmgtHao 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 ENi The geodesic equation is a generalization of the 11 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 1 x lEN 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 y ab A M C EN such that y tT 0 In simple terms we are asking for the curves which are as straight as possible subject to the constraint that they lie within Mi Example Suppose that M s zlmzn1 6 En 11 2W1 1 Let el and eg be two unitlength vectors in S which are perpendicular to each other and de ne the unitspeed great circle 7 z a b gt S by 7t cos te1 sin te2i Then a direct calculation shows that ry t 7coste1 7 sin te2 77ti Hence y tT 0 and 7 is a geodesic on Sni We will see later that all unit speed geodesics on S are obtained in this manner 132 Piecewise smooth paths For technical reasons it is often convenient to consider the more general space of piecewise smooth paths abMpq piecewise smooth paths 7 ab A M 7a 10712 4 an excellent exposition of which is given in Milnor 9 1li By piecewise smooth we mean 7 is continuous and there exist to lt t1 lt lt tN wit to a and tN I such that 7lti1ti is smooth for 1 S i S Ni In this case a van39atz39zm of 7 is a map A it 676 A 9abMpyq such that 10 7 and if 1 766 gtlt ab A M is de ned by ast dst then there exist to lt t1 lt lt tN with to a and tN I such that Miss X 2712 12 is smooth for 1 g i S Ni As before we nd that b 201 01 d1 ltJltaltsgtgtgt 0 lt07tgt ltotgtdt 5 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 If I Mil tLHETlLVO Mm 531th a short calculation shows that 16 yields the rst variation formula d E ltJltaltsgtgtgt 5 N71 7 Va made Z Wu Mm war 16 50 1 i1 whenever i is any variation of 7 with variation eld Vi If d WWW g 0018 0 50 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 7 ti 7ti7 for every i and T 0 Thus critical points on the more general space of piecewise smooth paths are also smooth geodesicsi Exercise I Due Friday April 8 Suppose that M2 is the right circular cylinder de ned by the equation 12 y2 l 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 7abt 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 13 We call M the con guration space of the simple mechanical systeml lf 7 ab A M represents the motion of the system 1 7 t t kinetic energy at timet 2 quotY quotY 7t potential energy at time t Example 1 the Kepler problem If a planet of mass m is moving around a star of mass M0 with M0 gtgt m the star assumed to be stationary we might take M R3 7 000 lt mdz dz dy dy d2 dz 7GmM MI y 2 OA 12 y2 22 Here G is the gravitational constant Sir Isaac Newton derived Kepler7s three laws from this simple mechanical system using his second law 17 Force massacceleration where Force grad i L8 The central symmetry of the problem suggests that we should use spherical co ordinatesi But that raises the question How do we express Newton7s second law in terms of general curvilinear coordinates Needless to say this system of equa tions can be solved exactly thereby deriving Kepler7s three laws of planetary motion This was one of the major successes of Sir Isaac Newton7s Philosophiae naturalis pn39ncz39pia mathematica which appeared in 1687 Example 2 the gravitational eld of an oblate ellipsoid The sun actually rotates once every 25 days and some stars actually rotate much more rapidly Thus the gravitational potential of a rapidly rotating star should differ somewhat from the simple formula given in Example 1 For the more general situation we let M R3 7 D where D is a closed compact domain with smooth boundary containing the origin lt mdz dz dy dy d2 d2 and 45 M A R is the Newtonian gravitational potential77 determined as follows We imagine rst that the mass say p0 were all concentrated at the point zoyo20 E Di Then just as in Example 1 we can set 7G Mr y 2 mp0 x1 7 WV y 7 yo 2 7 20 More generally if pzyz represents mass density at zyz E D the con tribution to the potential zyz from a small coordinate cube located at iii with sides of length di dg d is given by Tgmpi g 2 didgdz 172ye2z7 2 l4 Summing over all the in nitesimal contributions to the potential we nally obtain the desired formula for potential MW 2 Maw 19 D Iii2y2252 Remarkably if the mass distribution is spherically symmetric one gets exactly the same function as in 17 with M0 being the integral of p over Di But in the case of an oblate ellipsoid the potential is different One can do the integration numerically and determine the numerical solutions to Newton7s equations in this case using software such as Mathematica and one nds that the orbits are no longer exactly closed but the perihelion of the planet precesses insteadi More generally still one could consider a nebula in space within the region D represented by a dust with mass density pzyzi It turns out that the Newtonian gravitationl potential77 zyz de ned by 19 is the solution to Poissonls equation 2 2 2 9 lt15 9 lt15 9 lt15 w 67 Q 47erp which goes to zero like lT as one approaches in nity a fact which could be proven using the divergence theoremi Example 3 the rigid body To construct an interesting example in which the con guration space M is not Euclidean space we take M 503 the group of real orthogonal 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 gt Mi lfp is a point in the rigid body with coordinates 11 12 13 at time t 0 we suppose that the coordinates of this point at time t will be 11 aulttgt ma alga W 12 v where W and amt amt 6503 13 a31t am am and 70 I the identity matrix Then the velocity vt of the particle p at time t will be 1 ml Ella14ml vlttgt W as 221 0sz 13 211 03239 W and hence vt vt a2ita jtzizji 11 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 1 Z ltB pzl12zgzizjdzld12dzsgt aita jti 217 15 We can rewrite this as l V K E Zcija2ita j t where cij pzl12zgzlzjdzld12dzg am 3 and de ne a Riemannian metric on M 503 by WW Wt Z Ozjaia htj i 1 10 ijk Then once again l27 t 7 represents the kinetic energy this time of the rigid bo y when its motion is represented by the curve 7 z a gt 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 A R can represent the potential energy for the rigid body In classical mechanics books the motion of a top is described 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 This means that astrologers will have to relearn their craft every few thousand years because the sun will traverse a different path through the t quot 39 orcing a 39 39 of t 39 signsi We want a formulation of Newton7s second law 18 which helps to solve prob lems such as those we have just mentioned ln Lagrangian mec anics the equa tions of motion for a simple mechanical system 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 precisely for a simple mechanical system M q we de ne the Lagrangian as a function on the tangent bundle of 7 L TM A R by v vv 7 q57rv where 7r TM A M is the usual projectioni We can then de ne the action J19abMpyq HR by 1 JW v tgtgtdt As before we say that 7 E Q is a critical point for J if 13 holds We can then give the Lagrangian formulation of Newton7s law 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 in a Riemannian manifold from p to q of shortest length is put into the broader context of nding the trajectories for simple mechanical systems Although we will focus on the geodesic problem the earliest workers in the theory of geodesics must have been partially motivated by the fact that we were simply studying simple mechanical systems in which the potential function is zero e will soon see 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 is a coordinate system on M and we can express L in terms of the coordinates 11 i i i 1 1391 i i on 7r 1U described by 11 If 7t 11t i i i I t and MO 11tiHz ti1tuiint then 7 t zlti i i z ta c1tmi ti EulerLagrange Theorem A point 7 E QM M p q is a critical point for the action J 42gt its coordinate functions satisfy the Euler Lagrange equations 1 d 1 63517393 0 111 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 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 1tiu z t s nti Let We ddtxzpixty Then I Jds xit s it i i i x39it s iti 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 7 n d BL 1 b n d BL 0a g m ea 17 i b n a 39z39 gtwdta ga wdt and hence d E ltJltaltsgtgtgt 1 M d M sata gm wlll Thus if 7 is a critical point for J we must have 1 M d M Z 0a glazi7ltaiigtl dn for every choice of smooth functions In particular if 77 ab A R is a smooth function such that 77a077b7 77gt0 on tub and we set w W zilttgta39cilttgtgt 7 ltltzilttgt lttgtgtgtl 7 M d M 2 A quotlt0 81139 7 E dt0 Since the integrand is nonnegative it must vanish identically and hence 111 must 0 then For a simple mechanical system the EulerLagrange equations yield a derivation of Newton7s second law of motion Indeed if n lt7gt Z gijdzidzj ij1 then in the standard coordinates 11 i i 1 1391 i in quotYt Z gij117 71niiij 7 zliuz i ij1 Hence 8 71 n ngkjk 845 azi72vz hill 781 k1 aL V d 6 39 V n V V39J 7 ZJ39J39k x39J aw 29M dt aw 2 am I 2911 11 k1 11 where W d2zjdt2i Thus the EulerLagrange equations become n n n v 991 v 1 99k v 9 IV J k 7 7 j J k 7 29111 aku I 7 2 V hi I I hi 11 61 61 18 OI n v 1 n 991 agik 99k v 19 J 7 J 7 J J k 7 F29le 2 jg ark 811 81139 I I 811 We multiply through by the matrix 917 which is inverse to gij to obtain n n Bab l l wk 7 l 1 Tjksz fizgl 1 12 Jk1 11 where n 1 6939 69M 1991 pl lk J 7 J 1 13 1 2 g9 811 l 81 81k The expressions F j are called the Christo el symbolsi Note that if 11 i i I are rectangular cartesian coordinates in Euclidean space the Christoffel symbols vanishi We can interpret the two sides of 112 as follows n il Z F kijik accelerationl jgtk1 n h 7 39 l 7 i 1 g BIZ 7 force per unit mass Hence equation 112 is just the statement of Newton7s second law force equals mass times acceleration for simple mechanical systems In the case where 45 0 we obtain differential equations for the geodesics onM n Z rgkijik 0 114 jk1 where the Fjvk7s are the Christoffel symbolsi 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 XZMHTM suchthat onidM where 7r TM A M is the usual projection or equivalently a smooth map X M 4 TM such that Xp E TpMi The restriction of a vector eld to the domain U of a smooth coordinate system 11 i i i 7 I can be written as XlU Zfiaii where fizUARi i1 If we evaluate at a given point p E U this specializes to n E X i 7i i P 1 P 611 p A vector eld X can be regarded as a rstorder differential operator Thus if g M A R is a smooth function7 we can operate on g by X7 thereby obtaining a new smooth function Xg M A R by Xgp We let XM denote the space of all smooth vector elds on Mr It can be regarded as a real vector space or as an fMmodule where fM is the space of all smooth realvalued functions on M7 where the multiplication HM x XltMgt a 2W is de ned by fXp fpXp De nition A connection on the tangent bundle TM is an operator V XM gtlt XM a XM that satis es the following axioms where we write VXY for VXY fogyz fVXZgVyZ7 1 15 VZltfX9Y ZfXfVZXZ9YszK 116 for g 67M and XYZ E Note that 116 is the usual Leibniz rule77 for differentiation We often call VXY the covariant derivative of Y in the direction of i Lemma 1 Any connection V is local that is ifU is an open subset ofM XlU E 0 E VxYlU E 0 and VyXlU E 07 for any Y E Proof Let p be a point of U and choose a smooth function f M A R such that f E 0 on a neighborhood of p and f E 1 outside Ur Then XlU E 0 E fX E Xi Hence VXYP VfXYP fPVXYP 07 VYXP VYfXP fPVYXP YfPXP 0 20 Since p was an arbitrary point of U7 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 T i Thus we can restrict to the domain U ofa local coordinate system 117 i i 7 I and de ne the components 1 U A R of the connection by a n k a Z k1 Vii3x1 Then if X and Y are smooth vector elds on U7 say 16 Va XZfaxi wgwav i1 we can use the connection axioms and the components of the connection to calculate VXY a a Iii n n V 91 n i V vXYZ ijaxj k ijfjgk a 117 i1 j1 j 1 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 117 Because of the previous lemma7 we can VUX E TpM7 whenever v 6 TpM and X is a vector eld de ned along some curve tangent to v at p by setting va wixm for any choice of extensions 7 of v and X of Xi In particular7 if 7 db A M is a smooth curve7 we can de ne the vector eld qu along 7 Recall that we de ne the Lie bracket of two vector elds X and Y by X7 7 If X and Y are smooth vector elds on the domain U of local corrdinates 11 i i i 7 I say 16 Va Xf7 kgglw 1 then 7 n iagj 8 n jafi 8 lel 7 f Brig 71219 811 811 zj1 Fundamental Theorem of Riemannian Geometry If M7 lt is a Rie mannian manifold there is a unique connection on TM such that 21 l V is symmetric that is VXY 7 VyX X7Y for XY E XM 2 Vismetric that is XltYZgt ltny7ZgtltK VXzgt 1 0rX7Y7 Z 6 This connection is called the Levi Civita connection of the Riemannian manifold M lt3 gt To prove the theorem we express the two conditions in terms of local coordinates 11 l l l 1 defined on an open subset U of Ml In terms of the components of V de ned by the formula 6 7 k a Viv3117M ZFZjiaxk 118 k 1 the rst condition becomes 8 8 k 7 k 7 Tij 7 Fji s1nce 761 ifsz 7 Thus the Pgs are symmetric in the lower pair of indices If we write lt Z gijdzi dzj ij1 then the second condition yields agij 7 a a a 7 a a a a axle w WW t ltvvwwwgt n a a a n a n n l l l l ltZrkiw lt72Fwwgt Egl m 29139sz l1 l1 l1 l1 In fact7 the second condition is equivalent to 69 n n 76 Z gljr Z gilrijl 1 19 l1 l1 We can permute the indices in 1 197 obtaining a V n n Zglkrlj Zgjlrlk 1 20 l1 ll and n n a Zgur k ngr ii 1 21 l1 ll Subtracting 119 from the sum of 120 and 121 and using the symmetry of F j in the lower indices yields 691 69M agij n l 7 7 7 2 F vi 811 l 81 81k lgng 1 Thus if we let gij denote the matrix inverse to gij we nd that 1 n 99k 99m 99 z 7 7 lie J 7 J Fij 7 2 Jay 81139 l 811 81k 7 122 which is exactly the formula 113 we obtained before by means of Hamiltonls princip er This proves uniqueness of the connection which is both symmetric and met rici For existence we de ne the connection locally by 118 where the T jls are de ned by 122 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 EN and any vector eld Y on E can be written as i6 i N YZfazi where filE HR In this case the LeviCivita connection VE has components Pg 0 and there fore the operator VE satis es the formula 6 811 N V Y 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 VXYP VilaPDT 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 Mr Note that if 7 z a b A M is a smooth curve then VW W WWW 23 so a smooth curve in M Q EN is a geodesic if and only if Vqy E 0 If we want to develop the subject independent of Nash s imbedding theorem we can make the 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 LeviCivita connection In terms of local coordinates if n dwowa 7 dt aw then a straightforward calculation yields d2zioy dzjoydzkoy a V 7 l W Z a Z F dt dt aw 1 23 11 jk1 This reduces to the equation 114 we obtained before from Hamiltonls princi plei Note that 3 V WCW 2ltVWC W 0 so geodesics automatically have constant speed More generally if 7 ab A M is a smooth curve we call Vqy the accel eration of 7 Thus if M lt is a simple mechanical system its equations of motion can be written as Vw aw 124 where in terms of local coordinates 11 i i i I on U E M gram ZgWavazixaazj Note that these equations of motion can be written as follows Lac 7 It db 7 1 25 dquot 7 n l 39k n 1393 To Ejk1rjkrjz 2219 13 In terms of the local coordinates 11 i i i 1 1391 i i on 7r 1U g TM this system of differential equations correspond to the vector eld ia ivaab a XZI Z erkxjxki291 Bil i1 i1 jk1 It follows from the fundamental existence and uniqueness theorem from the theory of ordinary differential equations 1 Chapter IV 4 that given an 24 element 1 E TpM there is a unique solution to this system de ned for t E 76 e for some 6 gt 0 which satis es the initial conditions 9WD Elli W0 55111 In the special case where 45 0 we can restate this as Existence and Uniqueness Theorem for Geodesics Given p E M and v E TpM there is a unique geodesic 7 z 76 e A M for some 6 gt 0 such that W3 P and W0 v Simplest Example In the case of Euclidean space E with the standard Euclidean metric gij 61 the Christoffel symbols vanish l k 0 and the equations for geodesics become d2zi dt2 7 The solutions are I I I 11 alt 121 the straight lines parametrized with constant speed Exercise II Due Friday April 15 Consider the upper halfplane H2 zy E R2 z y gt 0 with Riemannian metric lt dz 8 dz dy dy the socalled Poincare upper half plane at Calculate the Christoffel symbols Pg bi Write down the equations for the geodesics obtaining two equations d2z 7 d2y 7 Wquotquot W cl Show that the vertical halflines z c are the images of geodesics and nd their constant speed parametrizationsi 16 First variation of J intrinsic version 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 lt7t7tgtdt has geodesics as its critical points and recast the argument in a form that is independent of choice of isometric imbeddingi 25 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 26 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 126 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 27 for a simple mechanical system M lt gt such as those considered in 77i We would nd after a short calculation that b b dJ7V ltvlttgt7ltvmlttgtgtdte d Vvtdt b 7 W WWgt6 i grad 7tgt th Just as in 1i4 the critical points are solutions to Newtonls equation 124 17 Lorentz manifolds The notion of Riemmannian manifold has a generalization Which is extremely useful in Einsteinls theory of relativity both special and general as described for example is the standard texts 10 or 13 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 lt 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 ltw 39gtp Z 9oltPgtdIllp drjlpv ij1 Where the functions gij U A R are smooth The conditions that lt 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 gt1o lt45 0 gt Iqu Where Ipxp and Iqxq are p X p and L 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 Ln 7 1 is called a Lorentz metrici A pseudoRiemannian manifold is a pair M lt Where M is a smooth manifold and lt is a pseudoRiemannian metric on Mi Similarly a Lorentz manifold is a pair M lt Where M is a smooth manifold and lt 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 127 is the elapsed time measured by a clock moving along the world line 7 We call L7 the proper time of 7 The Twin Paradox The fact that elapsed time is measured by the integral 127 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 29 128 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 para ox Equation 128 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 af ne parametrization such curves have zero acceleration In general relativity Minkowki spce time is replaced by a more general Lorentz manifold The world line of a massive body not subject to any forces other than gravity will also maximize L and if it is appropriately parametrized it will 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 J 9pm M p q A R de ned by JW lt7 t7 tgtdt The critical points of J for a Lorentz manifold M lt are called its 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 VXY 7 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 122 Finally by the rst variation formula one shows that a smooth parametrized curve 7 z a b gt M is a geodesic if and only if it satis es the equation Vqy E 0 We can now summarize the main ideas of general relativity that are treated in much more detail in 10 or 13 General relativity is a theory of the gravi tational force and there are two main components 1 The matter and energy in spacetime tells spacetime how to curve in ac cordance with the Einstein eld equations These Einstein eld equations are described in terms of the curvature of the Lorentz manifold and de termine the Lorentz metric We will describe curvature in the following sections 2i Timelike geodesics are exactly the world lines of massive objects which are subjected to no forces other than gravity while lightlike geodesics are the trajectories of light rays Riemannian geometry generalized to the case of Lorentz metrics was exactly the tool that Einstein needed to develop his theory 18 The RiemannChristoffel curvature tensor Let M lt be a Riemannian manifold or more generally a pseudoRiemannian manifold with LeviCivita connection Vi If XM denotes the space of smooth vector elds on M we de ne RXM gtlt XM gtlt XM 7 XM by RXYZ VXVyZ 7 VyVXZ 7 VXyZi We call R the RiemannChristo el curvature tensor of M lt Proposition 1 The operator R is multilinear over functions that is RfXYZ RX fYZ RXYfZ fRXYZi Proof We prove only the equality RXYfZ fRX YZ leaving the others as easy exercises RXYfZ vayfZ 7 Vyvxfz 7 VXyfZ VXYfZ fVYZ VYXfZfVXZ leYlfZ fVXYZ XYfZ YfVXZ XfVYz fVXVYZ YXfZ XfVYZ YfVXZ fVYVYZ leYlfZ fVXYZ fVvaZ7 VyVXZ 7VXyZ fRXYZi Since the connection V can be localized by Lemma 1 of 5 so can the curvature that is if U is an open subset of M RX YZlU depends only XlU YlU and ZlUi Thus Proposition 1 allows us to consider the curvature tensor as de ning a multilinear map szTprTprT7pM4TpMi Moreover the curvature tensor can be determined in local coordinates from its component functions Rijk U 7gt R de ned by the equations a a a n l a R w 31 Proposition 2 The components R jk of the Riemann Christoffel curvature tensor are determined from the Christoffel symbols F k by the equations a a n n Rim 0110 walk Z F39lmj 9 m1 m1 The proof is a straightforward computation From the RiemannChristoffel tensor one can in turn construct other local in variants of Riemannian geometry For example the Ricci curvature of a pseudo Riemannian manifold M lt is the bilinear form Ricp TpM gtlt TpM A R de ned by Ricpz y Trace of v gt gt Rpvzyi Of course the Ricci curvature also determines a bilinear map Ric XM gtlt XM In terms of coordinates we can write Ric Z Rijdri dz where 1 21 ij1 k1 Finally the scalar curvature of M lt is the function 8 M HR de ned by s Z ginZj ij1 where gij gij 1i It is easily veri ed that s is independent of choice of local coordinates The simplest example of course is Euclidean space ENi In this case the metric coef cients gij are constant and hence it follows from 122 that the Christoffel symbols 1quot 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 EN is given by the simple formula N a N a v 216161 Emma i1 i1 Similarly Minkowski spacetime Lnl which consists of the manifold Rn1 with coordinates t 11 i i I and Lorentz metric lt gt 752dt dt Zdzi dzi 11 has vanishing Christoffel symbols T2 0 and hence vanishing curvature 32 The next class of examples are the submanifolds of EN or submanifolds of Minkowski spacetime with the induced Riemannian metric It is often relatively easy to calculate the curvature of these submanifolds 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 v E TplENi lfp E M and v E TplEN we let v vT vt where vT E TpM and vLLTpMi 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 VEYWDT 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 2M gtlt 2M A XHM by aX Y v wm We call a the second fundamental form of M in ENi Proposition 3 The second fundamental form satis es the identities afXY aXfY faXY aXY aYXi Indeed aux Y vfxw fltv Ygti face Y X W V fYt XfY NEEV 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 aX Y 7 aY X v y 7 v Xi 0 There is some special terminology that is used in the case where 7 ab A M Q E is a unit speed curve In this case we say that the acceleration 7 t E T leN is the curvature of 7 while y tT Va7 t geodesic curvature of 7 at t y tL ay ty t normal curvature of y at t 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 EN is given by the Gauss equation ltRXYW Z aX Z aY W 7 aX W aY Z 129 where X Y Z and W are elements of XM and the dot on the right denotes the Euclidean metric in the ambient space E Proof Since Euclidean space has zero curvature v vgw 7 v v w 7 vamw 7 0 130 and hence if the dot denotes the Euclidean dot product 0 VEVEW Z 7 VEVEW Z7 VY1W Z 7 XV W Z 7 vgw v Z 7 YV W Z VEW vZ 7 vfbg w Z XltVyW Z 7 ltVyWVXZ 7 aY W aX Z YWXW Z WXWVYZ aX7W 107 Z ltVXYW Z Thus we nd that o 7 WWW 2gt 7 am W altX W 7 ViVXVV7 Z aX W aY W 7 ltVXylV Z This yields VXVyW7VyVXW7 VXyW Z aY W aX W7aX W aY W which is exactly 129 Example 1 We can consider the sphere of radius a about the origin in Enlz S a zlp l zn1 E En1 112 In12 a2 If y 76 e 7gt S a Q Enl is a unit speed great circle say 7t acoslate1 asinlate2 where e1e2 are orthonormal vectors located at the origin in EMA then a direct calculation shows t at 1 WW 7N7t7 where Np is the outward pointing unit normal to S a at the point p E S a In particular WWWW WWT 07 so 7 is a geodesic and in accordance with the Existence and Uniqueness The orem for Geodesics from 1l5 we see that the geodesics in S a are just the 34 constant speed great circlesi Moreover the second fundamental form of S a in En1 satis es 11 1 7Np for all unit length I E TpSnai If I does not have unit length then I z 1 1 M Mgt NltPgt e altrvrgt7ltzyzgtNltpgti By polarization we obtain 11 y 11 ygtNp for all Ly E TpSnai a Thus substitution into the Gauss equation yields 71 71 ltRltz7ygtw72gt lt7ltz72gtNltpgtgt lt7lty7wgtNltpgtgt 1lt N 1lt gtNltgt a 177 p a yvx p A Thus we nally obtain a formula for the curvature of Squot a ltRltzygtwzgt ltltz72gtltwgt 7 ltz7wgtltyzgtgt De nition If M1 lt and M2 lt are pseudoRiemannian manifolds a diffeomorphism 45 M1 A M2 is said to be an isometry i lt pv7 pwgt2 ltvwgt1 for all 1 w E Tle and all p E Mli 131 Of course we can rewrite 131 as gt2 lt gt1 where ltv7wgt2 45014107 pwgt27 for vvw E Tle Note that the isometries from a pseudoRiemannian manifold to itself form a group under compositioni Thus for example the orthogonal group On 1 acts as a group of isometries on S a a group of dimension 12nn 1 Just like we considered hypersurfaces in En1 we can calculate the curvature of spacelike hypersurfaces77 in Minkowski spacetime anJrli In this case the Christoffel symbols 1quot are zero so the LeviCivita connection VL of anl is de ned by a N 6 a N 6 v 1605 Xf gXf gt61 35 Suppose that M is an n dimensional manifold and L M 7gt anl is an imbeddingi We say that LM is a spacelz39ke hypersurftzce if the standard Lorentz metric on anll induces a positivede nite Riemannian metric on Mr For sim plicity let us set the speed of light 5 1 so that the Lorentz metric on an1 is simply n lt gtL 7dt dtZdzi 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 7gt XM on TM is given by the formula VXYP V YPT7 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 an1 by aXM gtlt XM7XLM by aXY xmy 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 ltRX7YW Zl ltaX7 Z7aY7 WgtL ltaX7 W7 107 ZgtL7 132 where X Y Z and W are elements of Example 2 We can now construct a second important example of an n dimensional Riemannian manifold for which we can easily calculate geodesics and curvature Namely we can set Hna tzl i i 1 6 anll t2 7 112 7 7 1 2 a2 t gt 0 the set of futurepointed timelike vectors V situated at the origin in anll such that V V 7a2 where the dot now denotes the Lorentz metric on anli Of course H a is nothing other than the upper sheet of a hyperboloid of two sheets Clearly H a is an imbedded submanifold of anJV1 and we claim that the induced metric on H a is positivede nite To prove this we could consider 11 i i i I as global coordinates on H a so that t a2 I12 Then 7 dt Ei1 zldzl a2 11 In2 and the induced metric on H a is E zizjdzi dzj 1 1 n n Wqde dz dz dzi lt39739gt 36 Thus I V E III a2 I12 In27 and from this expression we immediately see that the induced metric on H a is indeed positivede nite Just as S a is invariant under a large group of isometries so is H ai Indeed we can set gij 517 i 71 0 0 1 0 In I 0 0 1 and de ne a Lie subgroup 01 n of the general linear group by O1n A E GLn 1RATIlnA 1177 Elements of 01 n are called Lorentz transformations and it is easily checked that they act as isometries on anli The index two subgroup 01n A E O1n V futurepointing AV futurepointing preserves the upper sheet H a of the hyperboloid of two sheets and hence acts as isometries on H ai Of course just like On 1 the group 01n of orthochronous Lorentz transformations has dimension 12nn 1 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 eoi Using elementary linear algebra H must also contain a unit length spacelike vector e1 such that lte0e1gtL 0 In fact after an orthochronous Lorentz transformation we can arrange that 3 points along the taxis in anJrli Then the smooth curve 7 66 A an1 de ned by 7t acoshtae0 asinhtae1 lies in H a because a2coshta2 7 a2 sinhta2 a2i Note that 7 is spacelike and direct calculation shows that Wt WM 7ltsinhlttagtgt2 coma 1 Moreover Wt ltcoshlttagteo sinhlttagte1gt Ema where Np is the unit normal to H a at 11 Thus WWW WWW 07 37 so 7 is a geodesic and wowm WW gm Since 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 TplElna we have constructed all the unitspeed geodesics in H a Thus just as in the case of the sphere we can use the Gauss equation 13 to determine the curvature of H ai Thus the second fundamental form of H a in anl satis es az I 7Np for all unit length I E TpHna where Np is the futurepointing unit normal to Mr If I does not have unit length then I z 1 1 altmvmgtNp e azz7 ltzzgtNp 7 Z By polarization we obtain 11 y ltIygtNp for all z y E TpNnai Thus substitution into the Gauss equation yields ltRltzygtwzgt 7 lt ltz72gtNltpgtgt Gomw wgtNltpgtgt lt ltwgtNltpgtgt Since Np is timelike and hence ltNpNpgtL 71 we nally obtain a for mula for the curvature of S a ltRltzygtwzgt i21ltlt172gtlty7wgt 7 ltz7wgtltyzgtgt The Riemannian manifold H a is called hyperbolic space and its geome try is called hyperbolic geometry We have constructed a model for hyperbolic geometry the upper sheet of the hyperboloid of two sheets in Ln1 and have seen that the geodesics in this model are just the intersections with twoplanes passing through the origin in 1 i When M is either S a and H a 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 Hum all the postulates of Euclidean geometry are satis ed except for the parallel postulatei 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 lndeed7 the fol lowing proposition shows that a twodimensional Riemannian manifold has only one independent curvature component7 a threedimensional manifold only six7 a fourdimensional manifold only twenty Proposition 1 The curvature tensor R of a pseudo Riemannian manifold M7 lt satis es the identities l RXY 7RYX 2 RXYZ 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 129 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 index77 and write n P le ZylpRiZj 133 1 In terms of these components7 the curvature symmetries are P le lejiy P le leki Rlijk 07 P le Rklijy P le RijlkA In view of the last symmetry the lowering of the index into the third position in 133 is consistent with regarding the Rijlkls as the components of the map R TpM gtlt TpM gtlt TpM gtlt TpM A R by RXYZW ltRXYVVZgt 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 RX Y VXVY 7 VyVX 7VyVX 7 VXVy 7R0 X establishing the rst identityi Next RX7YZ RY7 ZXRZ7XY VXVyZ 7 VyVXZ VszX 7 VszX Vzva 7 vazY VxVyZ 7 VzY VyltVzX7 sz VzVXy 7 VyX 07 the last equality holding because V is symmetrici For the third identity7 we calculate ltRXYZZgt ltVXVyZZgt7 ltVyVXZZgt XltVyZZgt 7 ltVyZVXZgt 7 YltVXZ7 Zgt ltVXZVyZgt XYltZZgt 7 YXltZ7 Zgt XYltZ7 Zgt 0 Hence the symmetric part of the bilinear form WZ H ltRX7YW7Zgt is zero7 from which the third identity follows Finally7 it follows from the rst and second identities that ltRltX YW 2gt 7 7ltRltY XW 2gt 7 MM Wm 2gt ltRltW YgtX A and from the third and second that ltRltX7YgtW7 Zgt 7 7ltRltX7Ygt27 M 7 mm ZgtX M ltRltZ XgtY W Adding the last two expressions yields 2ltRX7YWZgt ltRX7WY7Zgt ltRltW YgtX 2gt ltRltY ZX Wgt ltRltz XgtK W 134 Exchanging the pair X7Y with W7 Z yields 2ltRW man 7 ltRltW XgtZYgt ltRX7 ZW7 Ygt ltRZ7 Y V7 X ltRY7 WZ7 Xgti 135 Each term on the right of 134 equals one of the terms on the right of 1357 so ltRX7YW7Zgt ltRW7ZX7Ygt7 nishing the proof of the proposition Using the rst and third of the symmetries we can de ne a linear map called the curvature aperatur7 R A2TpM 7gt A2TPM by y7 2 wgt ltRz 40 It follows from the fourth symmetry that R is symmetric UNI A y 2 A 10gt ltRz A wm A 9 so all the eigenvalues of R are real and AQTPM has an orthonormal basis con sisting of eigenvectorsi Proposition 2 Let RSTPMgtltTPMgtlt TpMXTpM7gtR be two quadrilinear functions which satisfy the curvature symmetries If Rzyzy Szyzy for all Ly E TpM then R 5 Proof Let T R 7 S Then T satis es the curvature symmetries and Tzyzy 0 for all Ly E TpMi Hence 0 Tzy2zyz Tz y z y Tz y z 2 Tz z z y Tz z z 2 2Tzyzz so Tzyzz 0 Similarly 0 Tz zyx 210 TIyzw Tz yzw 0 Tz wyzz w TIyzw Twy2zi Finally 0 21117972710 TQWJJU TWWJJ 2Tz y2w 7 Ty 21w 7 Tzz yw 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 ltR17 my 96gt lt17Igtltyyygt 7 my whenever zy is a basis for a The curvature symmetries imply that Ka is independent of the choice of basis KW Recall our key three examples the socalled spaces of constant curvature If M E then Ka E 0 for all twoplanes a Q TpMi If M S a then Ka E 1a2 for all twoplanes a g TpMi If M H a then Ka E ElaQ for all twoplanes a Q TpMi Along with the projective space lP a which is obtained from S a by identify ing antipodal points these are the most symmetric Riemannian manifolds possi ble it can be shown that they are the only n dimensional Riemannian manifolds which have an isometry group of maximal possible dimension 12nn 1 110 Gaussian curvature of surfaces We now make contact with the theory of surfaces in E3 as described in un dergraduate texts such as 12 If M 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 and is easily checked to also equal s2 where s is the scalar curvature of Mr As we saw in the previ ous section the Gaussian curvature of a twodimensional Riemannian manifold M determines the entire RiemannChristoffel curvature tensori An important special case is that of a twodimensional smooth surface M2 imbedded in E3 with M2 given the induced Riemannian metric We say that M is orientable if it is possible to choose a smooth unit normal N to M N M2 A S2 1 E E3 with Np perpendicular to TpMi Such a choice of unit normal is said to determine an orientation of M If NpM is the orthogonal complement to TpM in Euclidean space then the second fundamental form a TpM gtlt TpM A NpM determines a symmetric bilinear form h TpM gtlt TpM A R by the formula hzy azy Np for Ly 6 TpM and this Rvalued symmetric bilinear is also often called the second fundamental form of the surface Mi Note that if we reverse orientation h changes signi 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 8 8 Qij 7 for 17 172 If F M2 A E3 is the imbedding than the components of the induced Rieman nian metric also called the rst fundamental form are given by the formula 6 6 BF 8F 9174 42 Similarly we can de ne the components of the second fundamental form by 8 1quot hijhltgt fori7121 These components can be found by the explicit formula 9 92F 7 E 7 he VBarty 39N W39N Ifwe let a a X 7 Y 7 then it follows from the de nition of Gaussian curvature and the Gauss equation that ltPltX7YgtY7Xgt ltXXgtltYYgt 7 ltXYgt2 aX X aYY 7 aXY aXY 7 ltX7XgtltY7Ygt ltX7Ygt2 h11 h12 h h 7h h h 11 22 212 21 22 136 911922 f 912 911 912 921 922 In his General investigations regarding curved surfaces of 1827 Karl Friedrich Gauss de ned the Gaussian curvature by 1136 His Theorema Egregium stated that the Gaussian curvature depends only on the Riemannian metric gij From our viewpoint this follows from the fact that the RiemannChristoffel curvature is determined by the LeviCivita connection which in turn is determined by the Riemannian metric One says that the intrinsic geometry of a surface M2 Q E3 is the geometry of its rst fundamental form that is its Riemannian metric Everything that can be de ned in terms of the Riemannian metric such as the geodesics also belongs to the intrinsic geometry of the surface The second fundamental form a hN determines moreiit determines also the extrinisic geometry of the surface Thus for example a short calculation shows that the plane F11R2 A E3 de ned by F1uv u0v and the cylinder over the catenary F2 R2 A E3 de ned by F2uv lt10gltu W41 both induce the same Riemannian metric ltgtdu dudv dv 43 so they have the same intrinsic geometry yet their second fundamental forms are different so they have different extrinsic geometry If y z 76 e A S has unit speed then ay 0y 0 is simply the normal curvature of 7 at t 0 Thus the normal curvatures of curves passing through p at time 0 are just the values of the function H Tz M A R de ned by Hv avv where ngM is the unit circle in the tangent space Tz Mv Tpszv1i It is a well known fact from real analysis that a continuous function on a cir cle must achieve its maximum and minimum values These values are called the principal curvatures and are denoted by H1ltpgt and H2 The values of the principal curvatures can be found via the method of Lagrange multipliers from secondyear calculus one seeks the maximum and minimum values of the function 2 2 Hv Z hijvivj subject to the constraint ltvvgt Z gijvivj ll ij1 ij1 One thus nds that the principal curvatures are just the roots of the equation or h11 A911 h12 A912 0 h12 A921 h22 A922 One easily veri es that the Gaussian curvature is just the product of the prin cipal curvatures K l g but we can also construct an important extrinsic quantity 1 the mean curvature H H1 H2 Surfaces which locally minimize area can be shown to have mean curvature zeroi Such surfaces are called minimal surfaces and a vast literature is devoted to their study Example Let us consider the caterwz39d the submanifold of R3 de ned by the equatlon r xz2 y2 coshz where r 9 2 are cylindrical coordinates This is obtained by rotating the catenary around the zaxisi As parametrization we can take M R X 5391 and cosh u cos 1 FZRX51HE3 by Fuv coshusinv u Figure 11 The catenoid is the unique complete minimal surface of revolution in Here 1 is the coordinate on 51 Which is just the quotient group RZ Where Z is the cyclic group generated by 27L Then 6F sinhucosv 8F 7coshusinv 67 sinh u sin 1 and 67 cosh u cos 1 7 u 1 v 0 and hence the coef cients of the rst fundamental form in this case are 911 1 sinh2 u cosh2 u7 912 0 and 922 cosh2 u 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 6F 6F i sinhu cosv 7 coshusinv 7 coshu cosv 7 X 7 j sinh u sin 1 cosh u cos 1 7 cosh u sin 1 Bu 8v k cosh u smh u Thus a unit normal to S can be given by the formula Q X E 1 COSU N 73 a 7 7 sinv Q X Q coshu Bu 3v s1nhu 45 To calculate the second fundamental form7 we need the second order partial derivatives7 82F cosh u cos 1 82F 7 sinh u sin 1 72 cosh u sin 1 7 sinh u cos 1 7 Bu 0 9qu and 82F 7 coshucosv 7 cosh u sin 1 8112 These give the coefficients of the second fundamental form 62F 62F hnw39N 1v h12h21mN07 and 62F h22 W N1i From these components we can easily calculate the Gaussian curvature of the catenoi 1 K 77 cosh u4 Moreover7 one can check that the catenoid has mean curvature zero7 so it is a minimal surface In fact7 it is not difficult to show that the catenoid is the only complete minimal surface of revolution in Egi Exercise III Due Friday April 22 Consider the torus T2 6391 X 6391 with imbedding 2 cos u cos 1 F2T27gtS by Fu7v 2cosusinv 7 sin u Where u and v are the angular coordinates on the two 51 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 111 Matrix Lie groups In addition to the spaces of constant curvature7 there is another class of mani folds for Which the geodesics and curvature can be computed With relative ease7 46 the compact Lie groups with biinvariant Riemannian metricsi Before discussing these examples we give a brief review of Lie groups and Lie algebras For a more detailed discussion one could refer to Chapter 20 of Suppose that G is a Lie group and a 6 0 We can de ne the left translation by 0 LU CH C by LAT 0739 a map which is clearly a diffeomorphismi Similarly we can de ne right transla tion RU CH C by RAT Tar A vector eld X on G is said to be left invariant if LUX X for all a E G where LuXf Xf 0 Lu 0 1131 A straightforward calculation shows that if X and Y are left invariant vector elds on G then so is their bracket XYi Thus the space gXE XG LUX Xfor alla 6G is closed under Lie bracket and the real bilinear map 7 I g X g A g is skewsymmetric that is XY 7YX and satis es the Jacobi identity X7 Y7 le Y7 ZyXll Z X Yll 0 Thus g is a Lie algebra and we call it the Lie algebra of Cl If e is the identity of the Lie group restriction to TEG yields an isomorphism a g A TeGi The inverse T80 A 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 subgroups which are called matrix Lie groupsi For 1 S ij S n we can de ne coordinates GLnR 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 a z i39jk1 J If we allow X to vary over GLn R we obtain a left invariant vector eld which is de ned on GLnRl It is the unique left invariant vector eld on GLnR which satis es the condition 6 61 17 XAI Z a ij1 where I is the identity matrix the identity of the Lie group GLnRl Every left invariant vector eld on GLnR is obtained in this way for some choice of n X n matrix A A direct calculation yields XAXB XAB where A B AB 7 BA 13 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 TIC n X n matrices A with real entries with the usual bracket of matrices as Lie bracketl 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 3 t 9X 8 9X t for suf ciently small 8 and ti lndeed t gt gt 9Xst and t gt gt 9Xs 9Xt are two integral curves for X which agree when t 0 and hence must agree for all t From this one can easily argue that 9X t is de ned for all t E R and thus 9X provides a Lie group homomorphism 6X2RgtGl We call 9X the oneparameter group which corresponds to X 6 gr Since the vector eld X is left invariant the curve tgt gt LU6Xt 09xt R9Xz0 is the integral curve for X which passes through a at t 0 and therefore the oneparameter group 415 z t E R of diffeomorphisms on G which corresponds to X E g is given by 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 glnR we claim that the corresponding one parameter group is 1 1 6At e A ItA itQAQ Et3A3 i Indeed it is easy to prove directly that the power series converges for all t E R and termwise differentiation shows that it de nes a smooth map The usual proof that 61 6 65 extends to a proof that 6 SA ewe so t gt gt am is a oneparameter groupi Finally since d aw AetA e AA the curve t gt gt e A is tangent to A at the identity If G is a Lie subgroup of GLnR 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 GLnR ATA I Where denotes transpose and its identity component the special orthogonal group 50n A E 0n detA 1 In either case the corresponding Lie algebra is 0n A E glnlR em 6 0n for all t E R Differentiating the equation etATetA I yields etATATetA etATAetA 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 matrices The 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 TeG 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 GL2nRi The unitary group is Um A e ammo ATA I and its Lie algebra is 71 A E glnltC AT A 0 the Lie algebra of skewHermitian matrices while the special unitary group SUn A E Un detA 1 has Lie algebra 5un A E un TraceA 0 We can also develop a general linear group based upon the quaternionsi The space H of quaternions can be regarded as the space of complex 2 X 2 matrices of the form 7 t i2 1 iy Q 7 lt71iy tiizgt 138 where tzyz E R4 and i will As a real vector space H is generated by the four matrices 10 7 0 1 7 0239 7239 0 1Zlto 1gtv 1771 ogtv kt ogtv 0 727 the matrix product restricting to the cross product on the subspace spanned by i j and k Thus for example ij k in agreement with the cross product The conjugate of a quaternion Q de ned by 138 is if tiiz iziiy Q7ltziiy tizgt7 QTQ t2z2y2221 ltQ7QgtL where lt gt denotes the Euclidean dot product on Hi We can now de ne and GLnH n X n matrices A with quaternion entries det A y 0 the representation 1 38 showing how this can be regarded as a Lie subgroup of GL2nCi Finally we can de ne the compact symplectz39c group Spn A E GLnlF ATA I where the bar is now de ned to be quaternion conjugation of each quaternion entry of Al Note that Spn is a compact Lie subgroup of U2n and its Lie algebra is 5pn A E glnC AT A 0 where once again conjugation of A is understood to be quaternion conjugation of each matrix entryi Lie GroupLie algebra correspondence If G and H are Lie groups and h 39 A H is a Lie group homomorphism we can de ne a map Mtge by hX lheXel7 50 and one can check that this is yields a Lie algebra homomorphismi This gives rise to a covariant functor77 from the category of Lie groups and Lie group ho momorphisms to the category of Lie algebras and Lie algebra homomorphismsi A somewhat deeper theorem shows that for any Lie algebra g there is a unique simply connected Lie group G with Lie algebra gi For example there is a unique simply connected Lie group corresponding to 0n and this turns out to be a double cover of 50n called Spinni 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 A Lie algebra is said to be simple if it is nonabelian and has no nontrivial idealsi A compact Lie group is said to be simple if its Lie algebra is simple The compact simply connected Lie groups were classi ed by Wilhelm Killing and Elie Cartan in the late nineteenth century In addition to Spinn 5Un and 5pn there are exactly ve exceptional Lie groups The classi cation of these groups is one of the primary goals of a basic course in Lie group theory 112 Lie groups with biinvariant metrics It is easiest to compute curvature and geodesics in Riemannian manifolds which have a large group of isometries such as the space forms that we described before in li8i Compact Lie groups also have Riemannian metrics which have large isometry groupsi Indeed the Riemannian metric 110 used in classical mechanics for determining the motion of a rigid body is a leftinvariant metric on 503 that is the diffeomorphism LU 503 A 503 is an isometry for each a 6 503 as one checks by an easy calculation Even more symmetric are the biinvariant metrics which we study in this section De nition Suppose that G is a Lie group A pseudoRiemannian metric on G is biinvariant if the diffeomorphisms LT and RU are isometries for every 0 6 0 Proposition Every compact Lie group has a biinvariant Riemannian metric Proof We rst note that it is easy to construct leftinvariant Riemannian met rics on any Lie group Such a metric is de ned by a symmetric bilinear form on the Lie algebra ggtltggtRi If n dim G we can use a basis of left invariant oneforms 91 i i i 9 for G to construct a nonzero left invariant n form 9 91 97K This nowhere zero n form de nes an orientation for C so if G is compact we can de ne the HllllT integral of any smooth function f G A R by lcfe e Haar integral of fada f G G We can then average a given left invariant metric over right translations de ning ltlt gtgt g x g A R by ltltXYgtgt GltRXRYgtdai The resulting averaged metric is the soughtafter biinvariant metrici Example 1 We can de ne a Riemannian metric on GLnlR by n ltgt 2 dz mm 139 ij1 This is just the Euclidean metric that GLnlR inherits as an open subset of E Although this metric on GLnlR is not biinvariant7 we claim that the metric it induces on the subgroup 0n is biinvarianti To prove this7 it suf ces to show that the metric 139 is invariant under LA and RA when A E If A a E 0n and B E GLnlR7 then as o LAgtltBgt z ltABgt ZzzltAgtz ltBgt 2mm k1 k1 so that n oLA 61 It follows that n Ludzg Z aidzf 161 and hence LidJ Z Liar LZdI ij1 Z af x a dz Z aZaf z dz ijkl1 ijkl1 Since ATA I ELI alga 611 and hence L34 gt 2 5mm dag lt gt jkl1 By a quite similar computation7 one shows that RZlt lt gt for A E Hence the Riemannian metric de ned by 139 is indeed invariant under right and left translations by elements of the compact group Thus 139 in duces a biinvariant Riemannian metric on 0n7 as claimed Note that if we 52 identify TIOn With the Lie algebra 0n of skewsymmetric matrices this Rie mannian metric is given by ltXYgt TraceXTY for XY 6 GM Example 2 The unitary group Un is an imbedded subgroup of GL2nR Which lies inside 02n and hence if lt is the Euclidean metric induced on GL2n R Lid gt13 lt3 39gtE Rid gt137 for A E U01 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 ltX Y 2Re TraceXTl7 for X Y e um 140 Example 3 The compact symplectic group Spn is an imbedded subgroup of GL4n C Which lies inside 04n and hence if lt is the Euclidean metric induced on GL4nR L2lt lt BBQ gtE for A E Spni Thus the Euclidean metric on GL4n R induces a biinvariant Riemannian met ric on Spni Proposition Suppose that G is a Lie group With a biinvariant pseudo Riemannian metric lt Then 1 geodesics passing through the identity 6 E G are just the oneparameter subgroups of G 2 the LeVi CiVita connection on TC is de ned by VXY XY for XY E g 3 the curvature tensor is given by ltRXYVV Zgt ltXY Z Wgt for X Y Z W 6 g 141 Before proving this we need to some facts about the Lie bracket that are proven in 7 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 e ltlt tgtiltygtltpgtgt 70 1 42 53 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 The formula 14 for the Lie bracket has the following consequence needed in the proof of the theorem Lemma IfX is 3 Killing eld then ltVyX Z ltYVZXgt 0 for Y Z 6 Proof Note rst that if X is xed ltVYX7ZgtP and ltY7VZXgtP depend only on Yp and Zp Thus we can assume without loss of general ity that Zgt is constant Then since X is Killing ltq5tY is constant an zgt lt14 ltlt zgtvltzgtgt Hgt 7ltlX7Yl7Zgt 7ltY1lX7 o My t0 On the other hand since V is the LeviCivita connection 0 X04 Zgt ltVXYZgt mm Adding the last two equations yields the statement of the lemma Application le is a Killing eld on the pseudoRiemannian manifold M lt and 7 ab A M is a geodesic then since ltVyX Ygt 0 d 3mm W xgt NW xgt wavy0 o where we think of 7 as a vector eld de ned along 7 Thus 7 Xgt is constant along the geodesic This often gives very useful constraints on the qualitative behavior of geodesic How We now turn to the proof of the Proposition First note that since the metric lt is left invariant XY6g ltXYgt is constant Since the metric is right invariant each Rex is an isometry and hence X is a Killing eld Thus ltva Z ltvZX Y 0 for X Y Z 6 g 54 In particular 1 VXXYgt 7ltVyXXgt 7 YltXXgt 0 Thus VXX 0 for X E g and the integral curves of X must be geodesicsi Next note that 0 VxyX Y VYY ny VyXi Averaging the equations nyVYX0 ny7VyXXY yields the second assertion of the proposition Finally if XY Z 6 g use of the Jacobi identity yields RXYZ vxvyzi VyVXZ 7 VXyZ 1 1 1 1 1pc Y7 Z1 7 1w X7 2 7 given12 jHani On the other hand if XY Z 6 g 0 QXOC Zgt 2ltVXY7Zgt 2ltY7 VXZgt ltlX7Y17Zgt 07 X le Thus we conclude that 1 4 nishing the proof of the third assertioni ltRX7YWZgt 7 ltlX7YlyW17ZgtiltX7Y171Z7Wlgt7 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 isornetryi Indeed it is immediate that 1e 7id is an isometry and the identity 1 R071 01 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 L L071 0 1 0 L0 is an isometry Which reverses geodesics through 0 De nition A Riemanm39zm symmetric space is a Riemannian manifold M lt such that for each p 6 M there is an isometry Ip M gt M Which xes p and reverses geodesics through pi Examples include not just the Lie groups With biinvariant Riemannian metrics and the spaces of constant curvature but also complex projective space With the metric described in the next section 55 113 Projective spaces Grassmann manifolds We have jsut seen that if G is a compact Lie group with a biinvariant Rieman nian metric lt gt it is easy to compute the geodesics in G and the curvature of G The Riemannian symmetric spaces de ned at the end of the previous section provide a more general class of Riemannian manifolds n which one can easily calculate geodesics and curvature In 192627 Elie Cartan completely classi ed the Riemannian symmetric spaces the classi cation is presented in 5 and these provide a treasure box of examples on which one can test possible conjec turesi We give only the briefest introduction to this theory and consider a few symmetric spaces that can be realized as submanifolds M Q G with the metric induced from a biinvariant Riemannian metric on Gr An important case is the complex projective space with its Fubini Study metric a space which plays a central role in algebraic geometry e assume as known the basic theory of homogeneous spaces as described in Chapter 9 of 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 submersioni Moreover the map G X GH A GH de ned by 0 TH A 07H is smooth Suppose therefore that G is a compact Lie group with a biinvariant Rieman nian metric that s G A G is a group homomorphism such that 32 id and that HoEGsoo a compact Lie subgroup of G In this case the group homomorphism s induces a Lie algebra homomorphism s g A g such that 33 idi We let hXeg3XXv pX g3XXi Then g f 69 p 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 f is the Lie algebra of H and hence is isomorphic to the tangent space to H at the identity e while p is isomorphic to the tangent space to G H at eHi Under these conditions we can define a map F GH A G by FoH 03071 143 Note that ifh E H then Foh ohsh 1o l 030 1 so F is awell deflned map on the homogeneous space G Hi Lemma The map F de ned by 143 is an imbedding Moreover the geodesics for the induced Riemannian metric lt39739gtGHFlt39739gtG 1 44 56 on GH are just the left translates of one parameter subgroups ofG which are tangent to FGH at some point ofFGH Since GH is compact we need only show that F is a oneto one immersion But 08071 73771 42gt 7 10 37710 42gt 7 10 6 H so F is oneto onei To see that F is an immersion one rst checks that X E p 8671X67uxgt 8671Xgt8 7uxgt tgt gt 3e tX is a oneparameter group and checking the derivative at t 0 shows that 86quotX e X and hence Fe XH EQ Xi Thus F8H T8HGH E p a TEG 2g is injectivel Moreover Fae XH ae XsequotXsa 1 L0 0 R071e2tX for X E p 145 and since L0 and R071 are diffeomorphisms this quickly implies that FQUH is injective for each O39H E GH so F is indeed an immersion Thus FGH can be thought of as an imbedded submanifold of 0 The curves 7t Fae XH for a E G and X E p 146 are images under isometries of oneparameter subgroups of G by 145 and hence geodesics in G which lie within GHi For simplicity of notation we now identify GH with FGHi If Va and VGH are the LeviCivita connections on G and GH respectively and is the projection from TC to TGH then each curve 146 satis es T vSiwt vsWm o and hence is also a geodesic within GH verifying the last statement of the Lemmal The submanifolds H E G and GH intersect transversely at e 6 Ci We will see later that these submanifolds are in fact generated by the oneparameter subgroups 9X emanating from 6 within Q When X ranges over p the one parameter subgroups 9X cover H while when X ranges over p these one parameter subgroups cover GHl Finally note that the Lie group G acts as a group of isometries on GHi Moreover the isometry s of G takes GH to itself and restricts to 88H GH A GH de ned by seHaH saH an isometry which reverses geodesics through the point eHi Using the transitive group G of isometries one gets an isometry reversing geodesics through any point O39H in GH demonstrating that GH is indeed a Riemannian symmetric space 57 Example 1 Suppose G On and s is conjugation with the element 7I 0 pH lt 67X Iqxqgt where pq n Thus 3A 11731411737 for A E 0n7 and it is easily veri ed that s preserves the biinvariant metric and is a group homomorphism In this case H Op gtlt 0417 and the quotient OnOp gtlt 04 is the Grassmann manifold of real p planes in nspace The special case OnO1 gtlt On 7 1 of onedimensional subspaces of R is also known as real projective space RP Example 2 Suppose G Un and s is conjugation with the element 7I 0 I pxp gt where pqn 1711 lt 0 Iqxq In this case H Up gtlt Uq and the quotient UnUp gtlt Uq is the Grass mann manifold of complex p planes in n space he special case UnU1 gtlt Un 7 1 of complex onedimensional subspaces of C is also known as complex projective space Cpn l Example 3 Finally7 suppose G Spn and s is conjugation with the element 7 Ipo 0 7 Ipq7lt 0 Iqxq where pq7n In this case H Spp gtlt 5104 and the quotient SpnSpp gtlt 5104 is the Grassmann manifold of quaternionic p planes in n space The special case SpnSp1 gtlt Spn 7 1 of complex onedimensional subspaces of H is also known as quaternionic projective space Hpn Curvature Theorem Suppose that G is a compact Lie group With a biinvari ant Riemannian metric s G 7gt a Lie automorphism such that 32 id an that H a E G 8039 0 Then the curvature of the Riemannian metric on GH de ned by 144 is given by ltRltX 10W zgt ltX Y 2 W1 for Xx aw e TeasH e n 147 lndeed7 this curvature formula follows from the Gauss equation for a subman ifold M of a Riemannian manifold N7 lt when M is given the induced submanifold metric To prove this extended Gauss equation7 one follows the discussion already given in 187 except that we replace the ambient Euclidean space EN with a general Riemannian manifold N7 lt ThusiprMgNandvETpNwelet v UT vi where UT 6 TpM and viLTpM7 58 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 WWW VQYUUDT where VN is the LeviCivita connection on Ni If we let Xi 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 Yp 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 VfYVM the geodesic curvature of 7 in N while V 7fr Vf 7 geodesic curvature of 7 in M N L V77 With these preparations we can now state a77 normal curvature of 7 Extended Gauss Theorem The curvature tensor R of a submanifold M of a Riemannian manifold N with the induced Riemannian metric is given by the Gauss equation RM X YW Z RNX YW Z ltaX7Z7aY7WgtltaX7W7aY7Zgt7 148 where X Y Z and W are elements of The proof of the extended Gauss equation 148 is just like that of the previous Gauss equation except that we replace the equation 1 30 wit V V W 7 V V W 7 vfgmw ltRNXYW Z and then follow exactly the same steps as before Proof of the Curvature Theorem We simply note that since geodesics in GH are also geodesics in the ambient manifold G the second fundamental form a vanishes so 147 follows immediately from 148 Exercise IV Due Wednesday May 4 We consider the special case of the above construction in which G Un and s is conjugation with 71 0 I n 7 1 1 lt0 1ltn71gtxltn71gtgt 59 so that the fixed point set of the automorphism s is H U1 gtlt Un 7 1 and GH CP L li ai Recall that the Lie algebra un divides into a direct sum un 669p where fX gsXX7 pX gzs X7X7 and h is just the Lie algebra of Ul gtlt Un 7 1 Consider two elements 0 752 in 0 7772 7m X 2 0 1 0 and Y 77 9 1 0 5n 0 0 an 0 0 of p and determine their Lie bracket X7Y E SOLUTION To simplify notation7 we write Xltg 1 and Ylt2 17 149 where 5 and 77 are column vectors in Cn li Then ordinary matrix multiplication shows that ETU 0 TE 0 XY 7 YX 7 lt 0 EUT 0 in ianMTE 0 so that b Use the formula for curvature of GH to show that the sectional curvatures Ka for CP 1 satisfy the inequalities a2 g Ka S 4a2 for some a2 gt 0 SOLUTION As inner product on TUn7 we use ltABgt Re naceATB for AB 6 uni This differs by a constant fagtor from the Riemannian metric induced by the natural imbedding into Ea but with the rescaled metric ltX7 Ygt RGQTW when X and Y are given by 149 To simplify the calculations7 assume that ltXXgt ltYYgt 17 and ltXYgt 0 Then W an 1 and ET 7an 6 HR Q C 60 in other words 5T7 is purely imaginaryi Then lt1X7Y171X7Ylgt 71 777T55T 0 75T nT5 0 7 2Tracelt 0 i ET EnT lt 0 75771 g 7 1 4 ImQTmf 0 gt 7 7Tr 7 7 2 ace lt 0 WET WWW 775T 2 lImlt T gtl2 WW ImaTmf 15121n12 211me The last expression ranges between 1 and 4 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 Y ltX7Yllelegt K1 ltX7XgtltY7Ygt 7 ltX7Ygt2 7 2 15121n123llm5Tnl lies in the interval 1 4 achieving both extreme values when n 7 1 2 2 Remark Once one has complex projective space Cpn l with its Fubini Study metric the unique Riemannian metric up to scale factor which is invariant un der the action of one can construct a host of new examples of Riemannian manifolds namely the complex analytic submanifolds of Cpn li A famous the orem of Chow states these are algebraic varieties without singularities that is each such submanifold can be represented as the zero locus of a nite number of homogeneous polynomials with complex coef cients This brings us into contact with two major subjects within contemporary mathematics Kahler geometry and algebraic geometry over the complex eld both of which are treated in the beautiful text by Grif ths and Harris Chapter 2 Normal coordinates 21 De nition of normal coordinates 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 Such coordinates can also be constructed for pseudoRiemannian manifolds and their construction is based upon the following series of propositions Proposition 1 Suppose that M lt is a pseudo Riemannian manifold and p E M Then there is an open neighborhood V ofO in TpM such that ifv E TpM the unique geodesic 71 in M Which satis es the initial conditions 710 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 n dzj dzk F lv 77 7 0 Lit 392 1k dt dt k1 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 716 7196t 71 is de ned on 01 proving the proposition De nition The exponential map expp V A M is de ned by exppv 7v Remark Note that if G is a compact subgroup of GLn R With a biinvariant Riemannian metric lt as constructed in 1i12 eXpIA em for A E TIC E g g gnlRi Similarly if G is a compact subgroup of GLn R With a biinvariant Riemannian metric s G A G is a group homomorphism such that s2 id and that HUEGISUU we can divide the Lie algebra g of G into a direct sum g FJEB p Where h is the Lie algebra of H and p TEHGHi In this case the imbedding F GH A G F039H 08071 realizes GH as a submanifold of G and expeHA em for A E p E TeHGHi These facts explain the origin of the term exponential mapi77 Note that if v E V t gt gt expptv is a geodesic because expptv 7w 1 710 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 expp0 I To TpM TpM is an isomorphismi We identify To TPM With TpMi If v E TpM de ne Av R gt TpM by AUG tvi Then A 0 v and exppov exppoL0 expp 0 MW emmgt v7 t0 gmw 20 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 Within TM Which eXp maps diffeomorphically onto an open neighborhood W of 100100 in M X M 63 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 suf ces to show that exp0 is an epimorphismi Let nleXMHM 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 rrl o exp0 is an epimorphismi On the other hand the composition TPOMQTMEMXMEM is just exp7O and hence 71392 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 M lt is a Riemannian manifold and p 6 Mi If we choose a basis e1 i r en for T M orthonormal with respect to the inner product lt gtp we can de ne at coordinates I39lp 139 on TpM by n ai 42gt v Zaieii i1 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 zliuzn UAR by zioexpp 1 which we call Riemannz39an normal coordinates centered at p or sometimes sim ply normal coordinatesi lf Mlt is a Lorentz manifold as described in 1i7 which we study in units for which the speed of light is one we can choose linear coordinates 1390z391i i on TpM so that the Lorentz metric on TpM is represented by lt gt 7w 23 dabo Mail dab1 din dds 2 way my ij0 where 71 0 0 0 1 0 no 2 1 0 0 l These coordinates on TpM then project to coordinates 1011 l l I de ned on an open neighborhood U exppV ofp E M which we call Lorentz normal coordinatesl More generally we could de ne normal coordinates for a pseudoRiemannian manifold M lt of arbitrary signature Let us focus rst on the Riemannian case and suppose that in terms of Riemannian normal coordinates centered at p we have lt Z gijdzi dzjl ij1 It is interesting to determine the Taylor series expansion of the gijls about pi Of course we have gijp 61391quot To evaluate the rst order derivatives we note that whenever a1 l l a are constants the curve 7 de ned by Ii 0 7t at is a geodesic in M by de nition of the exponential mapl Thus the functions 11 II o 7 must satisfy the geodesic equation n 55k Z Piggy 0i ij1 Substitution into this equation yields n v Z Fgpala 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 119 that 691739 0 61k P Later we will see that the Taylor series for the Riemannian metric in normal coordinates centered at p is given by 1 n gij llj 7 g E Rik o kzl higher order terms kl1 This formula gives a very explicit formula for how much a Riemannian metric differs from the Euclidean metric near a given point p Note that when we look at a neighborhood of p under higher and higher magni cation it looks more and more like at Euclidean space the curvature measuring the deviation from atness There is a similar Taylor series for Lorentz manifolds 1 7 gij 7717 7 E Z Rikjlpzkzl h1gher order terms kl1 65 where the nifs are the components of the at Lorentz metric in Minkowski space time de ned by 21 In this case when we look at a neighborhood of an event p in a curved spacetime under higher and higher magni cation it looks more and more like at Minkowski spacetime in what is called an inertial framei77 These inertial frames represent coordinate systems in which gravitational forces vanish up to second order Thus a freely falling77 space station in outer space is in an innertial frame and for experiments inside the space station the gravitational force represented by the Christoffel symbols vanishesi The curvature tensor represents tidal forces that act over large distances They would become evident to an observer falling into the singularity in the middle of a black hole the tidal forces pulling an arm in one direction a leg in another until all classical forms of matter are destroyed Before establishing these Taylor series expansions we will need the socalled Gauss lemma 22 The Gauss Lemma Riemannian case Suppose that 11 i i i I are normal coordinates centered at a point p in a Riemannian manifold M lt and de ned on an open neigh borhood U of pi We can then de ne a radial function TzUHR by r zl2rn27 and a radial vector eld 5 on U 7 p by n Ii E S 7 7 81139 i For 1 S ij S n let Eij be the rotation vector eld on U de ned by E 8 l 7 J E z 61 I all 22 Lemma 1 EijS 0 Proof This can be veri ed by direct calculation For a more conceptual ar gument one can note that the oneparameter group of local diffeomorphisms 415 z t E R on U induced by Eij consists of rotations in terms of the normal coordinates so 41515 5 so Elj 7ltlt Lgtiltsgtgt 70 0 Lemma 2 IfV is the Levi Civita connection on M then V55 0 Proof If aq i i i a are real numbers such that 2ai2 1 then the curve 7 de ned by I I saw at 66 is an integral curve for 5 On the other hand W exp ltZait gt and hence 7 is a geodesic We conclude that all integral curves for 5 are geodesics and hence V55 0 Lemma 3 lt5 5 E 1 a 81139 Proof If 7 is as in the preceding lemma Wt7 Wt 2ltV7t7t77tgt 07 so 7 ltt must have constant length But W03 W0 201152 17 i1 so we conclude that lt5 5 E 1 Gauss Lemma I lt5 E E 0 Proof We calculate the derivative of lt5 Eijgt in the radial direction 5lt57Eijgt ltV557Eijgt lt57V5Eijgt lt57V5Eijgt 1 ltSVEWSgt EEijlt55gt 0 Thus lt5 Eijgt is constant along the geodesic rays emanating from 11 let M Then as 11 1 A lt00 llt57Eijgtl S llSllHEz jll llEijll A 0 If 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 Gauss Lemma II dT lt5 gt in other words dTltXgt lt5 X Whenever X is a smooth vector eld on U 7 p Proof It clearly suffices to prove this When either X 5 or X Eij In the rst case we so 1 ltSSgt while in the second dTEij EijT 0 lt57Eijl Lorentz case Suppose now that M7 lt is a Lorentz manifold with Lorentz normal coordinates IO 11 1 de ned on an open neighborhood U exppV oprM We let U exppv vzlt1vgtlt07 U exppv vzlt1vgtgt07 the images of the timelike and spacelike vectors in V respectively In this case7 we can de ne two functions MA A R by t mo 7 ml 7 e W rzUir HR by r 112zn27102 We now have two radial vector elds T on U and S on U de ned by 0 n n z E Ii E 10 E Ii E L7w27m 57Tazr i1 For 1 S ij S n let Eij be the rotation vector eld on U de ned by 22 and for 1 S i S n de ne the in nitesimal Lorentz transformation Em y Em 111 10 We can then carry out exactly the same steps for proving the Gauss Lemma Lemma 1 EijT Eij5 0 and EMT EOE5 0 Lemma 2 UV is the LeVi CiVita connection on M then VTT V55 0 Lemma 3 ltTTgt E 71 and 55 E 1 Gauss Lemma III For Lorentz manifolds ltT7Eijgt E 0 E ltT7E0igt and ltSinjgt E 0 E lt57Eoil The proofs are straightforward modi cations of the Riemannian case 23 Curvature in normal coordinates The following theorem explains how the curvature of a Riemannian manifold lt measures deviation from the Euclidean metric Taylor Series Theorem The Taylor series for the Riemannian metric gij in terms ofnormal coordinates centered at a point p E M is given by 1 n gij tilj 7 g E Rik o kzl higher order terms kl1 The Taylor series for a Lorentz metric gij in terms of normal coordinates centered at an event p E M is given 1 7 gij 7717 7 g E Rikjlpzkzl 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 n i a wZa axipi i1 Then the constant extension of w is the vector eld n 8 W Z i Z a 811 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 coordinates We de ne a quadrilinear map GszMXTpMXTprTpMAR 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 Gzy2 w Gyzzw 2 Gzy2 w Gzywz 3 Gzzzz 0 4 Gzzzy 0 5 Gzy2 w Cz w z y and 6 Gzy2 w Czzwy Gzwyz 0 69 Proof The second of these identities is immediate and the rst follows from equality of mixed partialsi The other identities require more work For the identity Cw7 w7 w7 w 07 we let W E aiBBzi then the curve 7 676 A M de ned by WWW W is an integral curve for W such that 70 p It is also a constant speed geodesic and hence WWltW Wgtp or We next check that Cw7 w7 w7 2 07 focusing rst on the Riemannian case 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 E E W E amp2 We consider the curve 7 in M de ned by 1107tt7 11107t07 forigtli Along 7 we have W S and Z lzlE127 so it follows from Gauss Lemma 1 that Zgt E 0 along 77 and hence VVVVWV7 Zgtp 0 23 The Lorentz case is similar We choose 7 to be spacelike or timelike7 so that 7 S or 7 T along 77 and to achieve 23 we need to apply Gauss Lemma III with two types of in nitesimal Lorentz transformations Eij and EM It follows from the rst four symmetries that whenever uv E TpM and t e R 0 Gu tv7u tv7utv7u7tv t3something t2Gv v u7 u 7 Cu7 u7 v 11 tsomething7 where we have used symmetries 1 and 2 to eliminate some terms in the sumi Since this identity must hold for all t7 the coef cient of it2 must be zero7 so wauw0mumw which yields the fth symmetryi To obtain the nal identity7 we let v1v2v3v4 e TpM and t1t2t3t4 e R and note that G Etivi Z tjvj Ztkvk Z tlvlgt 0i The coef cient of t1t2t3t4 must vanish7 and hence Z G UU17UU27UU3UU4gt 0A 0654 This together with the earlier symmetries7 yields the last symmetry 1quot Qide C y p Lemma RiljkltPgt Qikjl 7 9139ij 0 Proof Since the Christoffel symbols 1 vanish at p it follows that Now we let 8 7 1 p81 3 p7 811 i p7 81 a z 1 a E 091k 81139 ijp 7 281i 81k 811 7 BIZ ml 9 l 71 9 59 99 991k 9190 3 law f as all 10 and hence we conclude from Proposition 2 from 118 that 1 92911 92ij 929 929139 Rijlk p leijltpgt E 91er 7 aziazl l azjazl 7 azjazkl p 1 5 lgjlik Qik ijil mm P Qikjl 9139le 0 where the comma denotes differentiation and we have used the fth symmetry of G1 From the last lemma and the sixth symmetry7 we now conclude that RikjlltPgt RiljkP Qiljkp yijJMP 9139ij P i and 3gijklP We therefore conclude that BQgv 1 WW glRik W Riljkpl Substitution into the Taylor expansion 1 n 2917 k l 917 7 llj E 1621 azkazl pz z hlgher order terms 01 1 n 629139139 k l 91 7 771 E 16 axkaxl pz z hlgher order terms now yields the Taylor Series Theoremi 71 Exercise V Due Wednesday May 11 Suppose that is a two dimensional Riemannian manifold a Show that if 11 12 are Riemannian normal coordinates centered at p E M then 1 lt dzl2 d122 7 zldz2 712dzl2 higher order terms 24 where Kp is the Gaussian curvature of M at p b We can introduce geodesic polar coordinates 11 Tcost 127 sint9i Show that the formula 24 can be rewritten as 4 lt d7quot 8 dr 7 2 7 d9 8 dt higher order terms c If 7 gt 0 let Or be the geodesic circle de ned by the radial geodesic polar coordinate equal to Ti In the plane the length of this circle would be given by LCT 27ml Show that in the curved surface on the other hand we have Kp E11m 27 7 SHOT 7T THO 7 Thus in a surface with positive Gaussian curvature the length of the geodesic circle grows more slowly than in the Euclidean plane while when the Gaussian curvature is negative the length of the geodesic circle grows more rapidly Exercise VI Do not hand in Use normal coordinates to prove the Bianchi identity VX Ron ZgtWgt W PM XgtWgt w M YgtWgt 0 25 You can do this by rewriting 25 as RX Y VzW RY Z VxW RZ X VyW 0 Since 25 is a tensor equation it suf ces to prove this at any given point p 6 Mi But we can choose normal coordinates at p and let X Y Z and W be coordinate elds for the normal coordinatesi And then VzWp VXWp VyWp 0 making the identity triviali This illustrates the power of using normal coordi nates in calculations 24 Riemannian manifolds as metric spaces We can use the Riemannian normal coordinates constructed in the previous sections to establish the following important result Before stating it we note that a smooth curve A 01 A M is called regular if A t is never zero it is easy to prove that regular curve can always be reparametrized to have nonzero constant speed Local Minimization Theorem Suppose that M lt is a Riemannian manifold and that V 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 V and that y 01 A M is the geodesic de ned by 7t eprtv Let L eprv IfA 01 A M is any smooth curve With A0 p and A1 4 then 1 LA 2 LW and ifequality holds and A is regular then A is a reparametriza tion of 7 E JA 2 JW With equality holding only ifA 7 To prove the first of these assertions we use normal coordinates 11 i i I defined on U Note that LW Mg Suppose that A 01 A M is any smooth curve with A0 p and A1 4 Case I Suppose that A does not leave U Then it follows from Gauss Lemma 11 that LltAgt Wm WM HWMW 2 wt RltAlttgtgtdt 20 dTtdt 7 o A1 7 7 o A0 Lyi Moreover equality holds only if A t is a nonnegative multiple of RAt which holds only if A is a reparametrization of 7 Case II Suppose that A leaves U at some first time to E 0 1 Then Lo ltNlttgtAlttgtgtdtgt WW2 ltA lttgtRltAlttgtgtdt 2 0 WWW r e we 7 r 0 Mo e gt Lo The second assertion is proven in a similar fashion If M lt is a Riemannian manifold we can define a distance function dzMXMAR 73 by setting dp q inf L7 such that 7 01 A M is a smooth path with 70 P and 71 4 Then the previous theorem shos that dpq 0 implies that p qr Hence 1 dp q 2 0 with equality holding if and only if p q 2 d104 dqyp7 and 3 dmi S d104 dw 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 Mr In particular the distance function d M X M A R is continuous 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 7ap7 Yb and L7dP7 l 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 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 25 Completeness We return now to a variational problem that we considered earlieri Given two points p and q in a Riemannian manifold M does there exist a minimal geodesic from p to L For this variational problem to have a solution we need an hypothesis on the Riemannian metric De nition A pseudoRiemannian manifold M is said to be geodesically complete if geodesics in M can be extended inde nitely without running off the manifold Equivalently M 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 74 and the Grassmann manifolds On the other hand nonempty proper open subsets of any of these spaces are not geodesically complete Minimal Geodesic Theorem I Suppose that M lt is a connected and geodesically complete Riemannian manifold 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 av where v is a unit length vector in TpM which points in the direction77 of qi More precisely let Be 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 Mi Let 35 be the boundary of B5 and let S be the image of 539 under exppi 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 TpMi Finally we de ne 7 0a 7gt M by 7t expptvi Then 7 is a candidate for the minimal geodesic from p to q To finish the proof we need to show that 7a qr It will suf ce to show that d7t74 a 7 L 26 for all t E 0ai Note that d7tq 2 a 7 t because if d7tq lt a 7 t then WWI S 10770 d7t74 lt t a it a Moreover if 26 holds for to E 0 a it also holds for all t E 0 to because if t6 0t0 then d7t7 q S dWW Wo dWOoLq S to 7 t a n to a n t We let to supt E 0 a d7t q a 7 t and note that d7t0q 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 Local Mini mization Theorem from i dpq infdp39r d39rq 7 E S 6 infd7quotq 7 E S e dmq and hence a e dm q e d7e qi 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 dWOoMI infd7to L107 1 IT 6 5 6 77174 and hence a 7 to e dmq so a7 750 e dmqi Note that dpm 2 to 6 because otherwise 1074 S dmm 77174 lt t0 6 an to 6 a so the broken geodesic from p to 7t0 to m has length to e dp If the broken geodesic had a corner it could be shortened by rounding off the corner a fact which follows from the rst variation formula 16 for piecewise smooth pathsi Hence m must lie on the image of 7 so 7t0 e m contradicting the maximality of to It follows that to a d7aq 0 and 7a q nishing the proof of the theoremi Basic idea used in the preceding proof If you have a regular piecewise smooth curve that is 7t is never zero and at corners the left and right hand limits of 7 t exist and are nonzero then the curve can be shortened by rounding cornersi77 Needless to say this is a useful technique For a Riemannian manifold we also have a notion of completeness in terms of metric spaces Fortunately the two notions of completeness coincide HopfRinow Theorem Suppose that M is a connected Riemannian manifold Then M d is complete as a metric space if and only if M 139s geodesically complete To prove this theorem suppose rst that M is geodesically completei 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 We can assume without loss of generality that dqiqj lt e for some 6 gt 0 and let K dp 41 so that dpqi S K e for all ii It follows from the Minimal Geodesic Theorem that qi exppvi for some vi 6 TpM and S K er Completeness of R with its usual Euclidean metric implies that has a convergent subsequence which converges to some point 1 E TpMi Then 4 exppv is a limit of the Cauchy sequence qi and M d is indeed a complete metric space 0 prove the converse we suppose that M d is a complete metric space but Mn is not geodesically completei Then there is some unit speed geodesic 7 012 A M which extends to no interval 0b6 for 6 gt 0 Let be a sequence from 0 12 such that t gt bi lf pi 72 then dpipj g lti itjl so the sequence is a Cauchy sequence within the metric space M d Let p0 be the limit of Then by Corollary 4 from 2i1 we see that there is some xed 6 gt 0 such that expp1v is de ned for all lvl lt e wheni is suf ciently large This implies 7 can be extended a distance 6 beyond pi when i is sufficiently large yielding a contradiction 26 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 geodesicl Of course this can be thought of as a special case of periodic motion within classical mechanics Simplifying notation a little we let QMpq smooth maps 7 01 A M such that 70 p and 71 q and let 9Mpyq 7 6 9Mp741J7lt a Assuming that M is compact we can conclude from Proposition 3 of 211 that there is a 6 gt 0 such that any p and q in M with dp q lt 6 are connected by a unique minimal geodesic m I 0711 A M with LWM 41074 Moreover if 6 gt 0 is sufficiently small the ball of radius 6 about any point is geodesically convex and 7 depends smoothly on p and q If 7 z 12 12 5 A M is a smooth path and 62 e lt 27 then JW S a L S V2ae lt 6 a as we see from 112 Choose N E N such that 1N lt e and if 7 E 9M 1097 let pi 7iN for 0 S i S Ni Then 7 is approximated by the map 3 01 A M such that W map for te f Thus Ry lies in the space of broken geodesics77 BGNMpq maps 7 01 A M such that 7 2 171 is a constant speed geodesic and 9M p q is approximated by BGNMp74 7 6 BGNM P74 I JW lt a Suppose that 7 is an element of BGNM 104 1 Then if 39 2 Pi W 7 then dPzgt17Pi S M Fa lt Vgaf lt 57 77 so 3 is completely determined by PoyplpwpipprL where 100 10 PN qt Thus we have an injection N71 JIZBGNMP74aHMgtltMgtltgtltM jltvgtltvltagtwltgtgtv We also have a map 7 9M 104 gt BGNMpqa de ned as follows If 3 E QMpqa let T3 be the broken geodesic from 1 N71 PP0t0P17 N t0t0 PN7139Y T toq We can regard T3 as the closest approximation to 3 in the space of broken geodesics Minimal Geodesic Theorem II Suppose that M 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 it so that QMpqa is nonempty and let 3 be a sequence in QMpqa such that J3j A u Let 3139 7 j the corresponding broken geodesic from 1 N71 PPOthP1j7 N t0quot39t0 PN71j7 T tog 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 ii Hence a subsequence of converges to an element 3 E BCNMpqai Moreover 31 S limjaooJWj S limjaooJWj 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 li3i2i 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 BCNM 104 can be regarded as a finitedimensional manifold which approximates the infinitedimensional space 9M 104 This is a powerful idea which Marston Morse used in his critical point theory for geodesicsi See 9 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 VlzslaM and VQISIHM are freely homotopic if there is a continuous path I 01 X 5391 gt M such that F0t 71t and I 1t 72ti We say that M is simply connected if any smooth path 7 5391 A M is freely homotopic to a constant pathi Thus M is simply connected if and only if its fundamental group as defined in 4 is zero As before we can approximate the space MapSlM of smooth maps 7 z 5391 A M by a finitedimensional space where 5391 is regarded as the interval 01 with the points 0 and 1 identified This time the finitedimensional space is the space of broken geodesics77 BGN51M maps 7 01 A M such that 7 171 is a constant speed geodesic and 70 71 Just as before when a is sufficiently small then MaprM 7 e Maplt617Mgt JW lt a is approximated by BCN51Ma y E BCN51M Jy lt a Moreover if p 7iN then 7 is completely determined by 101710 1 1 710131pr Thus we have an injection N quot szGNSlM HMgtltMgtltgtltM 1 N 71 M i v N mm We also have a map 7 MapSlM A BGN51M defined as follows If 7 E MapSlMa let TM be the broken geodesic from 1 N71 70 to p1wltNgt to to PN7139YltTgt to PN Y1 79 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 notationl 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 u infJ3 3 e 7 Choose a gt u so that f 3 E f J3 lt a is nonempty and let 3 be a sequence in f such that J3j A Mi Let 3139 T3j the corresponding broken geodesic and from 1 N 71 PNj 70 t0 Pu 7 N to t0 PN71j 7 t0 PNj 71 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 it Hence a subsequence of converges to an element 3 E BGNSlMai Moreover JWJ S limjeooj j S limjeooJWj 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 51 M Finally 3 cannot have any corners because if it did we could decrease length by rounding cornersl This follows again from the rst variation formula for piecewise smooth curves given in ll3l2l 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 Remarks It was proven by Pet and Liusternik that any compact connected Riemannian manifold has at least one smooth closed geodesicl As in the proof of the preceding theorem one needs a constraint to pull against and such con straints are provided by the standard topological invariants of the free loop space MapSlM namely 7 Map 51 M and Hk MapSl M Z A nonconstant geodesic 3 z 51 A M is said to be prime if it is not of the form 3 0 7r where 7r 5391 A 51 is a covering of degree k 2 2i Wilhelm Klingenberg raised the question ls it true that any compact simply connected Rieman nian manifold has infinitely many prime smooth closed geodesics Gromoll and 80 Meyer 3 made a signi cant advance on this question by showing that if M had only nitely many prime geodesics then the ranks of HkMapSl M R must be unboundedi Unfortunately it is quite dif cult to calculate the integer cohomology ring of MapSlMi However Quillen and Sullivan were able to simplify the calculations of rational or real cohomology suf ciently via Sulli van s theory of minimal models to enable Vigu and Sullivan to show that if M has only nitely many prime geodesics then the real cohomology algebra H MapSl M R must be generated as an algebra by a single element Yet the basic question raised by Klingenberg still appears to be open for completely general compact simply connected Riemannian manifo s The partial solution of this question by Gromoll Meyer and others repre sents an impressive application of algebraic topology to a problem very much at the center of geometry and mechanics One need only recall that proving existence of periodic solutions to problems in celestial mechanics was one of the prime motivations for Henri Poincare7s research which led to the development of qualitative methods for solving differential equations 27 Parallel transport along curves Let M lt be a pseudoRiemannian manifold with LeviCivita connection Vi Suppose that 7 z a b A M is a smooth curve A smooth vector eld in M along 7 is a smooth function X z a b A TM such that Xt E TymM for all t E abi Note that we can de ne the covariant den39vatz39ve of such a vector eld along 7 VVX ab A TM WWXX 6 Tqui If 11 i i i I are local coordinates in terms of which 7 71 7 wt Z Li 7 0 71 i1 Xe 310 then a short calculation shows that vqXa Z View 2 F kltvlttgtgtdlzjdf7gtlttgtfklttgtl jk1 i1 ltrgti De nition We say that a vector eld X along 7 is parallel if VVX E 0 Proposition Ify ab A M is a smooth curve to 6 ab and v E T7tOM then there is a unique vector eld X along 7 Which is parallel along 7 and takes the value v at to VqX E 0 and Xt0 vi 2 81 Proof Suppose that in terms of local coordinates n i8 UZa 81139 i1 710 Then 27 is equivalent to the linear initial value problem dfi n 239 157 k7 i 7239 ding d f 70 worm It follows from the theory of ordinary differential equations that this initial value problem has a unique solution de ned on the interval a by If 7 z a b A M is a smooth path we can de ne a vector space isomorphism 7quot T7aM gt T7bM by 771 X02 where X is the unique vector eld along 7 which is parallel and satis es the initial condition Xa vi Similarly we can de ne such an isomorphism 7 if 7 is only piecewise smoothi We call 7 the parallel transport along i Note that if X and Y are parallel along 7 then since the LeviCivita con nection V is metric ltXYgt is constant along 7 indeed 7 ltX Y ltvXYgt ltXvYgt of It follows that 7 is an isometry from T7aM to T7bMi Parallel transport depends very much on the path 7 For example we could imagine parallel transport on the unit twosphere S2 Q E3 along the following piecewise smooth geodesic triangle 7 We start at the north pole n E S2 and follow the prime meridian to the equator then follow the equator through 9 radians of longitude and nally follow a meridian of constant longitude back up to the north pole The resulting isometry from TnS2 to itself is then just a rotation through the angle 9 28 Geodesics and curvature We now consider the differential equation that is generated when we have a deformation through geodesicsi77 Suppose that 7 z a b A M is a smooth curve and that a z 76 e X ab A M is a smooth map such that a0t 7ti We can consider the map a as de ning a family of smooth curves 13 a b A M for s E 76 e such that 10 y if we set dst 13 t A smooth vector eld in M along a is a smooth function X 7676 gtlt ab A TM such that Xst E TMSQM for all s t E 676 x a7bli We can take the covariant derivatives VaBSX and Wig31X of such a vector eld along a just as we did for vector elds along curves In fact7 we already carried out this construction in a special case in li6i lf 117 i i 7 I are local coordinates in terms of whic n a X87t 2101870 i1 Mm E 7 8a 7 n 8zioa E and we wr1te aha t 7 ghat 7 2787 611 t 7 then a short calculation yields 7 n Bfi n i 8zjoa k 8 Va35X37 75 i as jglmk 0 1716 87 75 w M5 if A similar local coordinate formula can be given for V3 atXi Of course7 important examples of vector elds along 1 include alandaia as at 7 and it follows quickly from the local coordinate formulae that 8 8 Vaas Vaaz Just as in 1i87 the covariant derivatives do not commute7 and this failure is described by the curvature Thus if X is a smooth vector eld along 1 8a 8a Viias 0 VaazX Vaaz 0 Vii35X R E7 a X We say that a a deformation of 7 and call 8a Xt E0175 6 T QM the corresponding deformation eldi Proposition 1 Ifa is a deformation such that each 13 is a geodesic then the deformation eld X must satisfy Jacobi s equation VqVqX RX77 0 2 8 83 Proof Since 13 is a geodesic for every 8 8a 8a Vaat E 0 and hence VaBSVBat 0 By the de nition of curvature see li8 8a 8a 8a 8a Vaanaas R 7 a E 0 8 8 8 8 0r Vaanaaz R 8 0 Evaluation at s 0 now yields 28 nishing the proof Remark The Jacobi equation can be regarded as the linearization of the geodesic equation near a given geodesic 7 De nition A vector eld X along a geodesic 7 which satis es the Jacobi equation 28 is called a Jacobi eldi Suppose that 7 is a unit speed geodesic and that E1i are parallel or thonormal vector elds along 7 such that E1 7 We can then de ne the component functions of the curvature with respect to E1 i by n REk 130E Z Rglei i1 where our convention is that the upper index 239 gets lowered to the third position If X E fiEi then the Jacobi equation becomes d2 n i j M ZRjuf 0 2 9 j1 This second order linear system of ordinary differential equations will possess a 2ndimensional vector space of solutions along 7 The Jacobi elds which vanish at a given point will form a linear subspace of dimension n Example Suppose that M lt is a complete Riemannian manifold of con stant sectional curvature kl Thus RXYW kltY WgtX 7 X In this case X will be a Jacobi eld if and only if V WV WX RX7 Y7 WWW WCWXl Equivalently if we assume that 7 is unit speed and write X E fiEi where E1i En is a parallel orthonormal frame along 7 such that E1 7 then dei 0 ddquot 7kfl forZSiSni 84 The solutions are dW and for 2 g i S n ai coslt bi sinlt for k gt 0 f t ai bit for k 0 210 ai cosh 7kt bisinhx7kt for k lt 0 Here a1 21 i i a b are constants of integration to be that are determined by the initial conditions De nition Suppose that 7 z a b A M is a geodesic in a pseudoRiemannian manifold with 7a p and 71 q We say that p and q are conjugate along 7 if p f q and there is a nonzero Jacobi eld X along 7 such that Xa 0 Xbi For example antipodal points on S are conjugate along the great circle geodesics which join them while E and Hquot do not have any conjugate pointsi Suppose that p is a point in a geodesically complete pseudoRiemannian manifold M and v E TpMi We can then de ne a geodesic 7v 01 A M by 71 t expptvi We say that 1 belongs to the conjugate locus in TpM if 71 0 and 71 l are conjugate along 71 Proposition 2 A vector v E TpM belongs to the conjugate locus if and only if epr is singular at 1 that is there is a nonzero vector w 6 T1 TpM such that exppw 0 Proof We use the following construction lf w 6 T1 TpM we de ne aw 66 gtlt 01 A M by awst expptv swi We set Bay as 0 A Jacobi eld along 71 which vanishes at 71 As w ranges throughout TpM Xw rnges throughout the ndimensional space of Jacobi elds along 71 which vanish at 71 Xwt 1 If exppw 0 then Xw is a Jacobi eld along 71 which vanishes at 710 and 71 1 so 1 belongs to the conjugate locusi 1 If 1 belongs to the conjugate locus there is a Jacobi eld along 71 which vanishes at 71 0 and 71 1 and this vector eld must be of the form Xw for some w 6 T1 TPM But then exppw Xw 1 0 and hence expp is singular at vi Example Let us consider the nsphere S of constant curvature one If p is the north pole in S it follows from 210 that the conjugate locus in TPS is a family of concentric spheres of radius kw where 85

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