Part III | Differential Geometry Definitions Based on lectures by J. A. Ross Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. This course is intended as an introduction to modern differential geometry. It can be taken with a view to further studies in Geometry and Topology and should also be suitable as a supplementary course if your main interests are, for instance in Analysis or Mathematical Physics. A tentative syllabus is as follows. • Local Analysis and Differential Manifolds. Definition and examples of manifolds, smooth maps. Tangent vectors and vector fields, tangent bundle. Geometric consequences of the implicit function theorem, submanifolds. Lie Groups. • Vector Bundles. Structure group. The example of Hopf bundle. Bundle mor- phisms and automorphisms. Exterior algebra of differential forms. Tensors. Symplectic forms. Orientability of manifolds. Partitions of unity and integration on manifolds, Stokes Theorem; de Rham cohomology. Lie derivative of tensors. Connections on vector bundles and covariant derivatives: covariant exterior derivative, curvature. Bianchi identity. • Riemannian Geometry. Connections on the tangent bundle, torsion. Bianchi's identities for torsion free connections. Riemannian metrics, Levi-Civita con- nection, Christoffel symbols, geodesics. Riemannian curvature tensor and its symmetries, second Bianchi identity, sectional curvatures. Pre-requisites An essential pre-requisite is a working knowledge of linear algebra (including bilinear forms) and multivariate calculus (e.g. differentiation and Taylor's theorem in several variables). Exposure to some of the ideas of classical differential geometry might also be useful. 1 Contents III Differential Geometry (Definitions) Contents 0 Introduction 3 1 Manifolds 4 1.1 Manifolds . .4 1.2 Smooth functions and derivatives . .5 1.3 Bump functions and partitions of unity . .5 1.4 Submanifolds . .6 2 Vector fields 7 2.1 The tangent bundle . .7 2.2 Flows . .8 2.3 Lie derivative . .8 3 Lie groups 10 4 Vector bundles 11 4.1 Tensors . 11 4.2 Vector bundles . 12 5 Differential forms and de Rham cohomology 15 5.1 Differential forms . 15 5.2 De Rham cohomology . 15 5.3 Homological algebra and Mayer-Vietoris theorem . 16 6 Integration 17 6.1 Orientation . 17 6.2 Integration . 17 6.3 Stokes Theorem . 18 7 De Rham's theorem* 20 8 Connections 21 8.1 Basic properties of connections . 21 8.2 Geodesics and parallel transport . 21 8.3 Riemannian connections . 22 8.4 Curvature . 22 2 0 Introduction III Differential Geometry (Definitions) 0 Introduction 3 1 Manifolds III Differential Geometry (Definitions) 1 Manifolds 1.1 Manifolds Definition (Chart). A chart (U; ') on a set M is a bijection ' : U ! '(U) ⊆ Rn, where U ⊆ M and '(U) is open. A chart (U; ') is centered at p for p 2 U if '(p) = 0. Definition (Smooth function). Let (U; ') be a chart on M and f : M ! R. We say f is smooth or C1 at p 2 U if f ◦ '−1 : '(U) ! R is smooth at '(p) in the usual sense. '−1 f Rn ⊇ '(U) U R Definition (Atlas). An atlas on a set M is a collection of charts f(Uα;'α)g on M such that S (i) M = α Uα. n (ii) For all α; β, we have 'α(Uα \Uβ) is open in R , and the transition function −1 'α ◦ 'β : 'β(Uα \ Uβ) ! 'α(Uα \ Uβ) is smooth (in the usual sense). Definition (Equivalent atlases). Two atlases A1 and A2 are equivalent if A1[A2 is an atlas. Definition (Differentiable structure). A differentiable structure on M is a choice of equivalence class of atlases. Definition (Manifold). A manifold is a set M with a choice of differentiable structure whose topology is (i) Hausdorff, i.e. for all x; y 2 M, there are open neighbourhoods Ux;Uy ⊆ M with x 2 Ux; y 2 Uy and Ux \ Uy = ;. (ii) Second countable, i.e. there exists a countable collection (Un)n2N of open sets in M such that for all V ⊆ M open, and p 2 V , there is some n such that p 2 Un ⊆ V . Definition (Local coordinates). Let M be a manifold, and ' : U ! '(U) a chart of M. We can write ' = (x1; ··· ; xn) where each xi : U ! R. We call these the local coordinates. Definition (Dimension). If p 2 M, we say M has dimension n at p if for one (thus all) charts ' : U ! Rm with p 2 U, we have m = n. We say M has dimension n if it has dimension n at all points. 4 1 Manifolds III Differential Geometry (Definitions) 1.2 Smooth functions and derivatives Definition (Smooth function). A function f : M ! N is smooth at a point p 2 M if there are charts (U; ') for M and (V; ξ) for N with p 2 U and f(p) 2 V such that ξ ◦ f ◦ '−1 : '(U) ! ξ(V ) is smooth at '(p). A function is smooth if it is smooth at all points p 2 M. A diffeomorphism is a smooth f with a smooth inverse. We write C1(M; N) for the space of smooth maps f : M ! N. We write C1(M) for C1(M; R), and this has the additional structure of an algebra, i.e. a vector space with multiplication. Definition (Curve). A curve is a smooth map I ! M, where I is a non-empty open interval. Definition (Derivation). A derivation on an open subset U ⊆ M at p 2 U is a linear map X : C1(U) ! R satisfying the Leibniz rule X(fg) = f(p)X(g) + g(p)X(f): Definition (Tangent space). Let p 2 U ⊆ M, where U is open. The tangent space of M at p is the vector space 1 TpM = f derivations on U at p g ≡ Derp(C (U)): The subscript p tells us the point at which we are taking the tangent space. Definition (Derivative). Suppose F 2 C1(M; N), say F (p) = q. We define DF jp : TpM ! TqN by DF jp(X)(g) = X(g ◦ F ) 1 for X 2 TpM and g 2 C (V ) with q 2 V ⊆ N. This is a linear map called the derivative of F at p. M F N g g◦F R Definition (Derivative). Let γ : R ! M be a smooth function. Then we write dγ (t) =γ _ (t) = Dγj (1): dt t @ n Definition ( ). Given a chart ' : U ! with ' = (x1; ··· ; xn), we define @xi R ! @ −1 @ = (D'jp) 2 TpM: @xi p @xi '(p) 1.3 Bump functions and partitions of unity Definition (Partition of unity). Let fUαg be an open cover of a manifold M.A 1 partition of unity subordinate to fUαg is a collection 'α 2 C (M; R) such that 5 1 Manifolds III Differential Geometry (Definitions) (i)0 ≤ 'α ≤ 1 (ii) supp('α) ⊆ Uα (iii) For all p 2 M, all but finitely many 'α(p) are zero. P (iv) α 'α = 1. 1.4 Submanifolds Definition (Embedded submanifold). Let M be a manifold with dim M = n, and S be a submanifold of M. We say S is an embedded submanifold if for all p 2 S, there are coordinates x1; ··· ; xn on some chart U ⊆ M containing p such that S \ U = fxk+1 = xk+2 = ··· = xn = 0g for some k. Such coordinates are known as slice coordinates for S. Definition (Immersed submanifold). Let S; M be manifolds, and ι : S,! M be a smooth injective map with Dιjp : TpS ! TpM injective for all p 2 S. Then we call (ι, S) an immersed submanifold. By abuse of notation, we identify S and ι(S). Definition (Regular value). Let F 2 C1(M; N) and c 2 N. Let S = F −1(c). We say c is a regular value if for all p 2 S, the map DF jp : TpM ! TcN is surjective. 6 2 Vector fields III Differential Geometry (Definitions) 2 Vector fields 2.1 The tangent bundle Definition (Vector field). A vector field on some U ⊆ M is a smooth map X : U ! TM such that for all p 2 U, we have X(p) 2 TpM: In other words, we have π ◦ X = id. Definition (Vect(U)). Let Vect(U) denote the set of all vector fields on U. Let X; Y 2 Vect(U), and f 2 C1(U). Then we can define (X + Y )(p) = X(p) + Y (p); (fX)(p) = f(p)X(p): Then we have X + Y; fX 2 Vect(U). So Vect(U) is a C1(U) module. Moreover, if V ⊆ U ⊆ M and X 2 Vect(U), then XjV 2 Vect(V ). Conversely, if fVig is a cover of U, and Xi 2 Vect(Vi) are such that they agree on intersections, then they patch together to give an element of Vect(U). So we say that Vect is a sheaf of C1(M) modules. Definition (Tangent bundle). Let M be a manifold, and [ TM = TpM: p2M There is a natural projection map π : TM ! M sending vp 2 TpM to p. Let x1; ··· ; xn be coordinates on a chart (U; '). Then for any p 2 U and vp 2 TpM, there are some α1; ··· ; αn 2 R such that n X @ vp = αi : @x i=1 i p This gives a bijection −1 n π (U) ! '(U) × R vp 7! (x1(p); ··· ; xn(p); α1; ··· ; αn); These charts make TM into a manifold of dimension 2 dim M, called the tangent bundle of M. Definition (F -related).