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Chapter 11 Riemannian Metrics, Riemannian Manifolds

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Chapter 11 Riemannian Metrics, Riemannian Manifolds 11.1 Frames Fortunately, the rich theory of vector spaces endowed with aEuclideaninnerproductcan,toagreatextent,belifted to the tangent bundle of a manifold. The idea is to equip the tangent space TpM at p to the manifold M with an inner product , p,insucha way that these inner products vary smoothlyh i as p varies on M. It is then possible to define the length of a curve segment on a M and to define the distance between two points on M. 541 542 CHAPTER 11. RIEMANNIAN METRICS, RIEMANNIAN MANIFOLDS The notion of local (and global) frame plays an important technical role. Definition 11.1. Let M be an n-dimensional smooth manifold. For any open subset, U M,ann-tuple of ✓ vector fields, (X1,...,Xn), over U is called a frame over U i↵(X1(p),...,Xn(p)) is a basis of the tangent space, T M,foreveryp U.IfU = M,thentheX are global p 2 i sections and (X1,...,Xn)iscalledaframe (of M). The notion of a frame is due to Elie´ Cartan who (after Darboux) made extensive use of them under the name of moving frame (and the moving frame method). Cartan’s terminology is intuitively clear: As a point, p, moves in U,theframe,(X1(p),...,Xn(p)), moves from fibre to fibre. Physicists refer to a frame as a choice of local gauge. 11.1. FRAMES 543 If dim(M)=n,thenforeverychart,(U,'), since 1 n d'− : R TpM is a bijection for every p U,the '(p) ! 2 n-tuple of vector fields, (X1,...,Xn), with 1 Xi(p)=d''−(p)(ei), is a frame of TM over U,where n (e1,...,en)isthecanonicalbasisofR .SeeFigure11.1. r e2 q p e1 φ φ -1 -1 φ (r) -1 U φ (q) -1 φ (p) Figure 11.1: A frame on S2. 544 CHAPTER 11. RIEMANNIAN METRICS, RIEMANNIAN MANIFOLDS The following proposition tells us when the tangent bun- dle is trivial (that is, isomorphic to the product, M Rn): ⇥ Proposition 11.1. The tangent bundle, TM, of a smooth n-dimensional manifold, M, is trivial i↵it possesses a frame of global sections (vector fields de- fined on M). As an illustration of Proposition 11.1 we can prove that the tangent bundle, TS1,ofthecircle,istrivial. 11.1. FRAMES 545 Indeed, we can find a section that is everywhere nonzero, i.e. anon-vanishingvectorfield,namely X(cos ✓, sin ✓)=( sin ✓, cos ✓). − The reader should try proving that TS3 is also trivial (use the quaternions). However, TS2 is nontrivial, although this not so easy to prove. More generally, it can be shown that TSn is nontrivial for all even n 2. It can even be shown that S1, S3 and S7 are the only≥ spheres whose tangent bundle is trivial. This is a rather deep theorem and its proof is hard. Remark: Amanifold,M,suchthatitstangentbundle, TM,istrivialiscalledparallelizable. We now define Riemannian metrics and Riemannian man- ifolds. 546 CHAPTER 11. RIEMANNIAN METRICS, RIEMANNIAN MANIFOLDS 11.2 Riemannian Metrics Definition 11.2. Given a smooth n-dimensional man- ifold, M,aRiemannian metric on M (or TM) is a family, ( , p)p M ,ofinnerproductsoneachtangent h i 2 space, TpM,suchthat , p depends smoothly on p, h i n which means that for every chart, '↵ : U↵ R ,for ! every frame, (X1,...,Xn), on U↵,themaps p X (p),X (p) ,pU , 1 i, j n 7! h i j ip 2 ↵ are smooth. A smooth manifold, M,withaRiemannian metric is called a Riemannian manifold. If dim(M)=n,thenforeverychart,(U,'), we have the 1 frame, (X1,...,Xn), over U,withXi(p)=d''−(p)(ei), n where (e1,...,en)isthecanonicalbasisofR .Sinceev- n ery vector field over U is a linear combination, i=1 fiXi, for some smooth functions, fi : U R,theconditionof Definition 11.2 is equivalent to the! fact that theP maps, 1 1 p d'− (e ),d'− (e ) ,pU, 1 i, j n, 7! h '(p) i '(p) j ip 2 are smooth. 11.2. RIEMANNIAN METRICS 547 If we let x = '(p), the above condition says that the maps, 1 1 x d'− (ei),d'− (ej) 1 ,x '(U), 1 i, j n, 7! h x x i'− (x) 2 are smooth. If M is a Riemannian manifold, the metric on TM is often denoted g =(gp)p M .Inachart,usinglocalcoordinates, we often use the notation2 g = g dx dx or simply ij ij i ⌦ j g = gijdxidxj,where ij P P @ @ gij(p)= , . @xi @xj *✓ ◆p ✓ ◆p+p 548 CHAPTER 11. RIEMANNIAN METRICS, RIEMANNIAN MANIFOLDS For every p U,thematrix,(gij(p)), is symmetric, pos- itive definite.2 The standard Euclidean metric on Rn,namely, g = dx2 + + dx2 , 1 ··· n makes Rn into a Riemannian manifold. Then, every submanifold, M,ofRn inherits a metric by restricting the Euclidean metric to M. n 1 For example, the sphere, S − ,inheritsametricthat n 1 makes S − into a Riemannian manifold. It is instructive to find the local expression of this metric for S2 in spherical coordinates. 11.2. RIEMANNIAN METRICS 549 We can parametrize the sphere S2 in terms of two angles ✓ (the colatitude)and' (the longitude)asfollows: x =sin✓ cos ' y =sin✓ sin ' z =cos✓. See Figure 11.2. z Θ φ y x Figure 11.2: The spherical coordinates of S2. 550 CHAPTER 11. RIEMANNIAN METRICS, RIEMANNIAN MANIFOLDS In order for the above to be a parametrization, we need to restrict its domain to V = (✓,') 0 <✓<⇡,0 < '<2⇡ . { | } Then the semicircle from the north pole to the south pole lying in the xz-plane is omitted from the sphere. In order to cover the whole sphere, we need another parametrization obained by choosing the axes in a suit- able fashion; for example, to omit the semicircle in the xy-plane from (0, 1, 0) to (0, 1, 0) and with x 0. − To compute the matrix giving the Riemannian metric in this chart, we need to compute a basis (u(✓,'),v(✓,')) 2 of the the tangent plane TpS at p =(sin✓ cos ', sin ✓ sin ', cos ✓). 11.2. RIEMANNIAN METRICS 551 We can use @p u(✓,')= =(cos✓ cos ', cos ✓ sin ', sin ✓) @✓ − @p v(✓,')= =( sin ✓ sin ', sin ✓ cos ', 0), @' − and we find that u(✓,'),u(✓,') =1 h i u(✓,'),v(✓,') =0 h i v(✓,'),v(✓,') =sin2 ✓, h i 2 so the metric on TpS w.r.t. the basis (u(✓,'),v(✓,')) is given by the matrix 10 g = . p 0sin2 ✓ ✓ ◆ 552 CHAPTER 11. RIEMANNIAN METRICS, RIEMANNIAN MANIFOLDS Thus, for any tangent vector w = au(✓,')+bv(✓,'), a, b R, 2 we have 2 2 2 gp(w, w)=a +sin ✓b . AnontrivialexampleofaRiemannianmanifoldisthe Poincar´eupper half-space,namely,theset H = (x, y) R2 y>0 equipped with the metric { 2 | } dx2 + dy2 g = . y2 11.2. RIEMANNIAN METRICS 553 Consider the Lie group SO(n). We know from Section 7.2 that its tangent space at the identity TISO(n)isthevectorspaceso(n)ofn n skew symmetric matrices, and that the tangent space⇥ TQSO(n)toSO(n)atQ is isomorphic to Qso(n)= QB B so(n) . { | 2 } If we give so(n)theinnerproduct B ,B =tr(B>B )= tr(B B ), h 1 2i 1 2 − 1 2 the inner product on TQSO(n)isgivenby QB ,QB =tr((QB )>QB )=tr(B>B ). h 1 2i 1 2 1 2 554 CHAPTER 11. RIEMANNIAN METRICS, RIEMANNIAN MANIFOLDS We will see in Chapter 13 that the length L(γ)ofthe curve segment γ from I to eB given by t etB (with 7! B so(n)) is given by 2 1 2 L(γ)= tr( B2) . − ✓ ◆ More generally, given any Lie group G,anyinnerproduct , on its Lie algebra g induces by left translation an h i inner product , g on TgG for every g G,andthis yields a Riemannianh i metric on G (which happens2 to be left-invariant; see Chapter 18). 11.2. RIEMANNIAN METRICS 555 Going back to the second example of Section 7.5, where we computed the di↵erential df R of the function f : SO(3) R given by ! 2 f(R)=(u>Rv) , we found that df (X)=2u>Xvu>Rv, X Rso(3). R 2 Since each tangent space TRSO(3) is a Euclidean space under the inner product defined above, by duality (see Proposition ?? applied to the pairing , ), there is auniquevectorY T SO(3) definingh thei linear form 2 R df R;thatis, Y,X = df (X), for all X T SO(3). h i R 2 R By definition, the vector Y is the gradient of f at R, denoted (grad(f))R. 556 CHAPTER 11. RIEMANNIAN METRICS, RIEMANNIAN MANIFOLDS We leave it as an exercise to prove that the gradient of f at R is given by (grad(f)) = u>RvR(R>uv> vu>R). R − More generally, the notion of gradient is defined as fol- lows. Definition 11.3. If (M, , )isasmoothmanifold h i with a Riemannian metric and if f : M R is a smooth function on M,thentheuniquesmoothvectorfieldgrad(! f) defined such that (grad(f)) ,u = df (u), h p ip p for all p M and all u T M 2 2 p is called the gradient of f. It is usually complicated to find the gradient of a function. 11.2. RIEMANNIAN METRICS 557 Awaytoobtainametriconamanifold,N,istopull- back the metric, g,onanothermanifold,M,alongalocal di↵eomorphism, ': N M. ! Recall that ' is a local di↵eomorphism i↵ d' : T N T M p p ! '(p) is a bijective linear map for every p N. 2 Given any metric g on M,if' is a local di↵eomorphism, we define the pull-back metric, '⇤g,onN induced by g as follows: For all p N,forallu, v T N, 2 2 p ('⇤g)p(u, v)=g'(p)(d'p(u),d'p(v)).
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