Class Note for MATH 538 with Professor Glickenstein at UA
Class Note for MATH 538 with Professor Glickenstein at UA
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Date Created: 02/06/15
Quick Introduction to Riemannian geometry David Glickenstein Math 538 Spring 2009 January 20 2009 1 Introduction We will try to get as quickly as possible to a point where we can do some geometric analysis on Riemannian spaces One should look at Tao s lecture 07 though I will not follow it too closely 2 Basics of tangent bundles and tensor bundles Recall that for a smooth manifold M7 the tangent bundle can be de ned in essentially 3 different ways Ui i are coordinates a TM X R N where for 3571 E Ui X R 3110 6 UJ X R we 239 have 711 N y w if and only iff y ti271 and w d ab271M v TpM paths 7 755 A M such that 70 p N where a N if qva o a 0 0 0 for every 239 such that p E Ui TM TpM pEM TpM to be the set of derivations of germs at 177 ie7 the set of linear funce tionals X on the germs at 17 such that X fg X fgp f pX g for germs fg at 17 TM TpM pEM On can de ne the cotangent bundle by essentially taking the dual of TpM at each point7 which we call Tpquot M7 and taking the disjoint union of these to get the cotangent bundle TM One could also use an analogue of the rst de nition7 where the only difference is that instead of using the vector space R one uses its dual and the equivalence takes into account that the dual space pulls back rather than pushes forward Both of these bundles are vector bundles One can also take a tensor bundle of two vector bundles by replacing the ber over a point by the tensor product of the bers over the same point7 eg7 TM TM TpM TpquotM pEM Note that there are canonical isomorphisms of tensor products of vector spaces7 such as V V is isomorphic to endomorphisms of V Note the difference between bilinear forms V 8 V7 endomorphisms V 8 V7 and bivectors V 8 V It is important to understand that these bundles are global objects7 but will often be considered in coordinates Given a coordinate ac and a point p in the coordinate patch7 there is a basis flp 7 7 lp for TPM and dual basis 111 p 7 7 dacan for Tg M The generalization of the rst de nition above gives the idea of how one considers the trivializations of the bundle in a coordinate patch7 and how the patches are linked together Speci cally7 if at and y give different coordinates7 for a point on the tensor bundle7 one has v a a a Z k 7 a b a Tag Cxaxi 39 w dx MewWm quot 3214 3yquot 8219 81 81b 8ng a a a TZJ k 7 4 da d5 d7 ab CMy 312 31 axkayaay 8w 8y 8yn 8y9 y 3 y yy where technically everything should be at p but as we shall see7 one can consider this for all points in the neighborhood and this is considered as an equation of sections Recall that a section of a bundle 7T E A B is a function f B A E such that 71 o f is the identity on the base manifold B A local section may only be de ned on an open set in B On the tangent space7 sections are called vector elds and on the cotangent space7 sections are called forms or leforms On a tensor bundle7 sections are called tensors Note that the set of form a basis for the vector elds in the coordinate x7 and 1302 form a basis for the local leforms in the coordinates Sections in general are often written as I or as C if we are considering smooth sections Now the equation above makes sense as an equation of tensors sections of a tensor bundle Often7 a tensor will be denoted as simply Ty Note that if we change coordinates7 we have a different representation Ti 2 of the same tensor The two are related by 8y4 By 8319 81a 81b 81a 9 V T52 7 Ti 506 21 One can also take subsets or quotients of a tensor bundle ln particular7 we may consider the set of symmetric 27tensors or antiesymmetric tensors sec tions of this bundle are called differential forms In particular7 we have the Riemannian metric tensor De nition 1 A Riemannian metric g is o twoitensor ie7 a section of TM TM which is I symmetric ie7 gX7Y g Y7X for all X7Y E TpM7 and I positive de nite ie g XX 2 0 all X E TPM and g XX 0 if and only if X 0 Often we will denote the metric as gij which is shorthand for gijdacidacj where dxidzj 100239 1307 dacj daci Note that ifgij 67 the Kronecker delta then iiidacidxj dacl2 dxn2 One can invariantly de ne a trace of an endomorphism trace of a matrix which is independent of the coordinate change since n a7 aazaay gTa ZT aya a ZTgag a ZTg a In fact for any complicated tensor one can take the trace in one up index and one down index This is called contraction Usually when there is a repeated index of one up and one down we do not write the sum This is called Einstein summation convention The above sum would be written T5 T3 It is understood that this is an equation of functions We cannot contract two indices up or two indices down since this is not independent of coordinate change try it However now that we have the Riemannian metric we can use it to lower an index and then trace so we get Tabgba T In order to raise the index we need the dual to the Riemannian metric which is gab de ned such that gabgbc 6 so gab is the inverse matrix of gab Then we can use gab to raise indices and contract if necessary Occasionally extended Einstein convention is used where all repeated indices are summed with the understanding that the Riemannian metric is used to raise or lower indices when necessary eg Tea Tabgab Since often we will be changing the Riemannian metric it becomes important to understand that the metric is there when extended Einstein is used 3 Connections and covariant derivatives 31 What is a connection A covariant derivative is a particular way of differentiating vector elds Why do we need a new way to differentiate vector elds Here is the idea Suppose we want to give a notion of parallel vectors In R we know that if we take vector elds with constant coe eicients those vectors are parallel at different 8 8 8 8 pomts That 1s the vectors wlw oer Elam and wlu iler lminare parallel In fact we could say that the vector eld 2 is parallel since vectors at any two points are parallel One might say it is because the coe eicients of the vector eld are constant not functions of 1 and 12 However this notion is not invariant under a change of coordinates Suppose we consider the new coordinates 311312 lt11122 away from 2 0 where it is not a diffeomorphism Then the vector eld in the new coordinates is 2 8y 8 2 i 3 21in i 4y73 8x1 8312 812 81 8311 8312 8311 8312 The coe eicients are not constant but the vector eld should still be parallel we have only changed coordinates so it is the same vector eld So we need a notion of parallel vector eld that is independent of coordinate changes or covariant Remember that we want to generalize the notion that a vector eld has constant coe eicients Let X Xiai be a vector eld in a coordinate patch Roughly speaking we want to generalize the notion that 88 0 for all 239 and j The problem occurred because gr y is different in different coordinates Thus we need to specify what this is Certainly since is a basis we must get a linear combination of these so we take a k a Viw 27 for some functions Pg These symbols are called Christoffel symbols To make sense on a vector eld we must have v X v Xjgt 7an a J k a 7 830i W an a J k 7 835 X rgt WC Notice the Leibniz rule product rule One can now de ne V for any vector Y W 33 by VyX VWLX Yi vix 6 This action is called the covariant derivative One now de nes If in such a way that the covariant derivative transforms appropriately under change of coordinates This gives a global object called a connection The connection can be de ned axiomatically as follows De nition 2 A connection on a vector bundle E A B is a map VI TB I E HFE X45 4 VX satisfying Tensoriality ie C B7linear in the rst component ie foyq5 foq5 Vygb for any function f and vector elds X Y I Derivation in the second component ie VX X q5 fVXqi I Rilinear in the second component ie VX 11 w aVX VX for a E R We will consider connections primarily on the tangent bundle and tensor bundles Note that a connection V on TM induces connections on all tensor bundles also denoted V in the following way a For a function f and vector eld X we de ne VXf Xf For vector elds X Y and dual form to we use the product rule to derive VX w YD X w YD wa Y w VXY and thus wa Y X w YD w VXY In particular the Christoffel symbols for the connection on TM are the negative of the Christoffel symbols of TM ie VLdaj il Zvkdxk 051 where If are the Christoffel symbols for the connection V on TM a For a tensor product one de nes the connection using the product rule eg VX Y w VxY wY wa for vector elds XY and leform w Remark 3 The Christo el symbols are not tensors Note that if we change coordinates from at to a we have a 7 855 a 7 855 a 82k 855 a ij 3 g aw 326 away 326 Bari aw 3556 which means that F2 may of 835 of 8ka J p 8352 81 835m 835281 815 One nal comment Recall that we motivated the connection by considering parallel vector elds The connection gives us a way of taking a vector at a point and translating it along a curve so that the induced vector eld along the curve is parallel ie Van 0 along y This is called parallel translation Parallel vector elds allow one to rewrite derivatives in coordinates that is if X Xi 36 is parallel then 8X i 817 ixkrglk 32 Torsion compatibility with the metric and LeviCivita connection There is a unique metric associated with the Riemannian metric called the Riemannian connection or LevieCivita connection It satis es two properties Torsionefree also called symmetric Compatible with the metric Compatibility with the metric is the easy one to understand We want the connection to behave well with respect to differentiating orthogonal vector elds Being compatible with the metric is the same as VX g Y Z gVXYZ gY VXZ Note that normally there would be an extra term ng Y Z so compatie bility with the metric means that this term is zero ie Vg 0 where g is considered as a 27tensor Torsion free means that the torsion tensor 739 given by T XY VXY 7 VyX 7 XY vanishes One can check that this is a tensor by verifying that 739 fXY 739 X fY f7 XY for any function It is easy to see that in coordinates the torsion tensor is given by 739le Fle F292 which indicates why torsionefree is also called symmetric Tao gives a short motivation for the concept of torsionefree Consider an in nitesimal parallelogram in the plane consisting of a point at the ow of as along a vector eld V to a point we will call at tV the ow of X along a vector eld W to a point we will call at tlV and then a fourth point which we will reach in two ways 1 go to at tV and then ow along the parallel translation of W for a distance i and 2 go to at lW and then ow along the parallel translation of V for a distance t Note that using method 1 we get that the point is 8 actVsWlS0ti as at tV 5W 0 t3 30 V 0 t3 at tV tW 7 tZVinr igik 0 t3 1 8 actVtWt27 as 30 Note that using method 2 we get instead V a 2 z j k 3 xtVtW7t WV Fjiaxk Ot Thus this vector is xt V W up to 0 t3 only in T fj Doing this around every in nitesimal parallelogram gives the equivalence of these two viewpoints Here is another Proposition 4 A connection is torsionifree if and only iffor any point p E M there are coordinates at around 17 such that If p 0 Proof Suppose one can always nd coordinates such that If p 0 Then clearly at that point Ti 0 However since the torsion is a tensor we can calculate it in any coordinate so at each point we have that the torsion vanishes Now suppose the torsion tensor vanishes and let an be a coordinate around 17 Consider the new coordinates T q xi q 3W1 F kp Hm T7 17 30k a i 3639 19 Then notice that 855i V V 6 Pie p 6 x5 7 x5 22 Pie p ask 7 ask 22gt 6 and so BM 352 i axj P 6quot Thus a is a coordinate patch in some neighborhood of 17 Moreover we have that 2 V 8 552 The P One can now verify that at p r pfwga aiifk 7W aifk J p 8352 81 835m 835281 815 Fi39cj P Fi39cj P I The Riemannian connection is the unique connection which is both torsione free and compatible with the metric One can use these two properties to derive a formula for it In coordinates one nds that the Riemannian connection has the following Christoffel symbols 1 8 8 8 Pg i9 835ng 92 927 One can easily verify that this connection has the properties expressed Note that the gjg in the formula etc are not the tensors but the functions This is not a tensor equation since If is not a tensor Also note that it is very important that this is an expression in coordinates ie that 0 33 Higher derivatives of functions and tensors One of the important reasons for having a connection is it allows us to take higher derivatives Note that one can take the derivative of a function without a connection and it is de ned as df Vf dfX foXf df Baggy One can also raise the index to get the gradient which is aiafna 27 835i 7 g Baci gmdf vif However to take the next derivative one needs a connection The second derivative or Hessian of a function is Hessf V2f Vdf V2f Vidf dxi v gag dxi 7 82f j 8f 7 8352317 dag 7 rg39kdxk dxi 7 82f 8f 7 812817 7 Pg dacj dxi Often one will write the Hessian as 32f 3f 2 7 V 7 7 7 k 7 Vijf 7 vzvjf 7 8352317 Fij Back Note that if the connection is symmetric then the Hessian of a function is symmetric in the usual sense The trace of the Hessian Af g ngf is called the Laplacian and we will use it quite a bit We also may use the connection to compute acceleration of a curve The velocity of a curve is quoty which does not need a connection but to compute the acceleration Vafy we need the connection one also sometimes sees the equivalent notation Diydt A curve with zero acceleration is called a geodesic Finally given any tensor T one can use the connection to form a new tensor VT which has an extra down index 4 Curvature One can de ne the curvature of any connection on a bundle E A B in the following way RI TM I TM I E HIKE RX7Yltl5 VXVY Vvalt15 Von1amp5 We will consider the curvature of the Riemannian connection on the tangent bundle One can easily see that in coordinates the curvature is a tensor denoted 8 8 8 Z 7k vjviiiR 7 vi vj 835 W 815 which gives us that 8 8 8 8 8 8 8 8 v r47 7v r57 7T4 7 rlrmii 7T5 fir irmi More Jlt more 8x2 1 811 1 Maw 811 C ax C aw a e a e m e m e a nk rm ijrim Fikrjm W So the curvature tensor is i e i 835i M 817 Often we will lower the index and consider instead the curvature tensor Rijk File FirHim F r m Rim RZLkgmz The Riemannian curvature tensor has the following symmetries Rijkg iRjikg iRijgk ngij These imply that R can be viewed as a selfeadjoint symmetric operator mapping 27forms to 27forms if one raises the rst two or last two indices I Algebraic Bianchi Rijkg Rjkig Rkijg 0 I Differential Bianchi ViRjkgm Vijigm kaijgm Remark 5 The tensor Rijkg can also be written as a tensor RXY Z W which is a function when vector elds XYZW are plugged in We will sometimes refer to this tensor as Rm The tensor Rfjk is usually denoted by RXY Z which is a vector eld when vector elds XY Z are plugged in Remark 6 Sometimes the up index is lowered into the 3rd spot instead of the 4th This will change the de nitions of Ricci and sectional curvature below but the sectional curvature of the sphere should always be positive and the Ricci curvature of the sphere should be positive de nite Remark 7 Note that If involved rst derivatives of the metric so Riemannian curvature tensor involves rst and second derivatives of the metric From these one can derive all the curvatures we Will need De nition 8 The Ricci curvature tensor Rij is de ned as Rij R5 RZijmggm Note that Rij RJi by the symmetries of the curvature tensor Ricci will sometimes be denoted Rc g or Rc X Y De nition 9 The scalar curvature R is the function R ginij De nition 10 The sectional curvature of a plane spanned by vectors X and Y is given by Roamx K0 gltX7XgtgltY7Ygt agony Here are some facts about the curvatures Proposition 11 1 The sectional curvatures determine the entire curvature tensor ie if one can calculate all sectional curvatures then one can calculate the entire tensor 2 The sectional curvature K XY is the Gaussian curvature of the surface generated by geodesics in the plane spanned by XY 5 The Ricci curvature can be written as an average of sectional curvature 4 The scalar curvature can be written as an average of Ricci curvatures 5 The scalar curvature essentially gives the di erence between the volumes of small metric balls and the volumes of Euclidean balls of the same radius 6 In 2 dimensions each curvature determines the others 7 In 5 dimensions scalar curvature does not determine Ricci but Ricci does determine the curvature tensor 8 In dimensions larger than 5 Ricci does not determine the curvature tensor there is an additional piece called the Weyl tensor 10 With this in mind we can talk about several different kinds of nonnegative curvature De nition 12 Let at be a point on a Riemannian manifold Mg Then at has 1 nonnegative scalar curvature ifR 2 0 2 nonnegative Ricci curvature at at if RC XX 13inin 2 0 for every vector X 6 TIM 5 nonnegative sectional curvature if RXYYX g R XY YX 2 0 for all vectors XY 6 TI M 4 nonnegative Riemann curvature or nonnegative curvature operator if Rm 2 0 as a quadratic form on 92M ie if Rijkgwijwkg 2 0 for all 27forms w wijdaci dacj where the raised indices are done using the metric g It is not too hard to see that 4 implies 3 implies 2 implies 1 Also in 3 dimensions 3 and 4 are equivalent ln dimension 4 and higher these are all distinct 11
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