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Lectures on Differential Geometry Math 240C

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Lectures on Differential Geometry Math 240C Lectures on Differential Geometry Math 240C John Douglas Moore Department of Mathematics University of California Santa Barbara, CA, USA 93106 e-mail: [email protected] June 6, 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. The starred sections are less central to the course, and may be omitted by some readers. i Contents 1 Riemannian geometry 1 1.1 Review of tangent and cotangent spaces . 1 1.2 Riemannian metrics . 4 1.3 Geodesics . 8 1.3.1 Smooth paths . 8 1.3.2 Piecewise smooth paths . 12 1.4 Hamilton's principle* . 13 1.5 The Levi-Civita connection . 19 1.6 First variation of J: intrinsic version . 25 1.7 Lorentz manifolds . 28 1.8 The Riemann-Christoffel curvature tensor . 31 1.9 Curvature symmetries; sectional curvature . 39 1.10 Gaussian curvature of surfaces . 42 1.11 Matrix Lie groups . 48 1.12 Lie groups with biinvariant metrics . 52 1.13 Projective spaces; Grassmann manifolds . 57 2 Normal coordinates 64 2.1 Definition of normal coordinates . 64 2.2 The Gauss Lemma . 68 2.3 Curvature in normal coordinates . 70 2.4 Tensor analysis . 75 2.5 Riemannian manifolds as metric spaces . 84 2.6 Completeness . 86 2.7 Smooth closed geodesics . 88 3 Curvature and topology 94 3.1 Overview . 94 3.2 Parallel transport along curves . 96 3.3 Geodesics and curvature . 97 3.4 The Hadamard-Cartan Theorem . 101 3.5 The fundamental group* . 104 3.5.1 Definition of the fundamental group* . 104 3.5.2 Homotopy lifting* . 106 ii 3.5.3 Universal covers* . 109 3.6 Uniqueness of simply connected space forms . 111 3.7 Non simply connected space forms . 113 3.8 Second variation of action . 115 3.9 Myers' Theorem . 118 3.10 Synge's Theorem . 120 4 Cartan's method of moving frames 123 4.1 An easy method for calculating curvature . 124 4.2 The curvature of a surface . 128 4.3 The Gauss-Bonnet formula for surfaces . 132 4.4 Application to hyperbolic geometry* . 136 5 Appendix: General relativity* 142 Bibliography 148 iii Chapter 1 Riemannian geometry 1.1 Review of tangent and cotangent spaces We will assume some familiarity with the theory of smooth manifolds, as pre- sented, for example, in [3] or in [12]. Suppose that M is a smooth manifold and p 2 M, and that F(p) denotes the space of pairs (U; f), where U is an open subset of M containing p and f : U ! R is a smooth function. If φ = (x1; : : : ; xn): U ! Rn is a smooth coordinate system on M with p 2 U, and (U; f) 2 F(p), we define @ −1 i (f) = Di(f ◦ φ )(φ(p)) 2 R; @x p where Di denotes differentiation with respect to the i-th component. We thereby obtain an -linear map R @ i : F(p) −! R; @x p called a directional derivative operator, which satisfies the Leibniz rule, ! ! @ @ @ i (fg) = i (f) g(p) + f(p) i (g) ; @x p @x p @x p and in addition depends only on the \germ" of f at p, @ @ f ≡ g on some neighborhood of p ) (f) = (g): @xi p @xi p The set of all linear combinations n X i @ a @xi i=1 p 1 of these basis vectors comprises the tangent space to M at p and is denoted by 1 n TpM. Thus for any given smooth coordinate system (x ; : : : ; x ) on M, we have a corresponding basis ! @ @ 1 ;:::; n @x p @x p for the tangent space TpM. The notation we have adopted makes it easy to see how the components (ai) of a tangent vector transform under change of coordinates. If = (y1; : : : ; yn) is a second smooth coordinate system on M, the new basis vectors are related to the old by the chain rule, n j j @ X @x @ @x j −1 = (p) ; where (p) = Di(x ◦ )( (p)): @yi @yi @xj @yi p j=1 p The disjoint union of all of the tangent spaces forms the tangent bundle [ TM = fTpM : p 2 Mg; 1 n which has a projection π : TM ! M defined by π(TpM) = p. If φ = (x ; : : : ; x ) is a coordinate system on U ⊂ M, we can define a corresponding coordinate system φ~ = (x1; : : : ; xn; x_ 1;:::; x_ n) on π−1(U) ⊂ TM by letting 0 1 0 1 n n i X j @ i i X j @ i x @ a A = x (p); x_ @ a A = a : (1.1) @xj @xj j=1 p j=1 p For the various choices of charts (U; φ), the corresponding charts (π−1(U); φ~) form an atlas making TM into a smooth manifold of dimension 2n, as you saw in Math 240A. ∗ The cotangent space to M at p is simply the dual space Tp M to TpM. Thus ∗ an element of Tp M is simply a linear functional α : TpM −! R: Corresponding to the basis ! @ @ 1 ;:::; n @x p @x p of TpM is the dual basis ! ( 1 n i @ i 1; if i = j, dx jp; : : : ; dx jp ; defined by dx jp j = δj = @x p 0; if i 6= j. 2 ∗ The elements of Tp M, called cotangent vectors, are just the linear combinations of these basis vectors n X i aidx jp i=1 Once again, under change of coordinates the basis elements transform by the chain rule, n X @yi dyij = (p)dxjj : p @xj p j=1 An important example of cotangent vector is the differential of a function at a point. If p 2 U and f : U ! R is a smooth function, then the differential of ∗ 1 n f at p is the element dfjp 2 Tp M defined by dfjp(v) = v(f). If (x ; : : : ; x ) is a smooth coordinate system defined on U, then n X @f dfj = (p)dxij : p @xi p i=1 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 = fTp M : p 2 Mg; 1 n which has a projection π : TM ! M defined by π(TpM) = p. If φ = (x ; : : : ; x ) is a coordinate system on U ⊂ M, we can define a corresponding coordinate system ~ 1 n −1 φ = (x ; : : : ; x ; p1; : : : ; pn) on π (U) ⊂ TM by letting 0 1 0 1 n n i X j @ i X j x @ a A = x (p); pi @ ajdx jpA = ai: @xj j=1 p j=1 (It is customary to use (p1; : : : ; pn) to denote momentum coordinates on the cotangent space.) For the various choices of charts (U; φ) on M, the corre- sponding charts (π−1(U); φ~) on T ∗M form an atlas making T ∗M into a smooth manifold of dimension 2n. We can generalize this construction and consider tensor products of tangent and cotangent spaces. For example, the tensor product of the cotangent space 2 ∗ with itself, denoted by ⊗ Tp M, is the linear space of bilinear maps g : TpM × TpM −! R: If φ = (x1; : : : ; xn): U ! Rn is a smooth coordinate system on M with p 2 U, we can define i j dx jp ⊗ dx jp : TpM × TpM −! R ! i j @ @ i j by dx jp ⊗ dx jp k ; l = δkδl : @x p @x p 3 Then i j dx jp ⊗ dx jp : 1 ≤ i ≤ n; 1 ≤ j ≤ n 2 ∗ 2 ∗ is a basis for ⊗ Tp M, and a typical element of ⊗ Tp M can be written as n X i j gij(p)dx jp ⊗ dx jp; i;j=1 where the gij(p)'s are elements of R. 1.2 Riemannian metrics Definition. Let M be a smooth manifold. A Riemannian metric on M is a function which assigns to each p 2 M a (positive-definite) inner product h·; ·ip on TpM which \varies smoothly" with p 2 M.A Riemannian manifold is a pair (M; h·; ·i) consisting of a smooth manifold M together with a Riemannian metric h·; ·i on M. Of course, we have to explain what we mean by \vary smoothly." This is most easily done in terms of local coordinates. If φ = (x1; : : : ; xn): U ! Rn is a smooth coordinate system on M, then for each choice of p 2 U, we can write n X i j h·; ·ip = gij(p)dx jp ⊗ dx jp: i;j=1 We thus obtain functions gij : U ! R, and we say that h·; ·ip 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 φ = (x1; : : : ; xn). Note that the functions gij satisfy the symmetry condition gij = gji and the condition that the matrix (gij) be positive definite. We will sometimes write n X i j h·; ·i|U = gijdx ⊗ dx : i;j=1 If = (y1; : : : ; yn) is a second smooth coordinate system on V ⊆ M, with n X i j h·; ·i|V = hijdy ⊗ dy ; i;j=1 it follows from the chain rule that, on U \ V , n X @yk @yl g = h : ij kl @xi @xj k:l=1 4 We will sometimes adopt the Einstein summation convention and leave out the summation sign: @yk @yl g = h : ij kl @xi @xj We remark in passing that this is how a \covariant tensor field 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.
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