Imperial College QFFF MSc ||||||{ Differential Geometry 2015-16 Lecture Notes ||||||{ Chris Hull Oct 2015 Contents 1 Introduction 5 1.1 Heuristic definition of (real) manifold . 5 1.2 Example: Stereographic projections of a 2-sphere; the mani- fold S2 ............................... 8 1.3 Open sets in Rm in the `usual topology' . 10 1.4 Topological space . 12 1.5 Topological structure of an m-dimensional real manifold M . 14 1.6 Example: circle S1 ........................ 15 1.7 Topological invariants and compactness . 18 1.8 Formal definition of smooth m-dim'l real manifold . 21 2 1.9 Example: real projective space RP . 23 1.10 Product manifold . 27 1.11 Example: Torus . 28 1.12 Complex manifold . 29 1.13 Example: S2 as a complex manifold . 30 n 1.14 Example: CP complex projective space . 31 1.15 Examples of not being a manifold . 33 1.16 Manifolds with boundary . 34 1 2 Differential maps, tangent vectors and tensors 36 2.1 Differentiable maps . 36 2.2 Embedding and submanifolds . 38 2.3 An example: CY 3 ......................... 39 2.4 Diffeomorphism . 40 2.5 Functions . 41 2.6 Curves . 42 2.7 Tangent vectors . 43 2.8 An aside: Dual vector spaces . 46 2.9 Cotangent vectors . 48 2.10 Tensors . 49 2.11 Contraction of tensors . 50 2.12 Tangent vectors act on functions . 51 2.13 Directional derivatives . 53 2.14 Differentials as Cotangent Vectors . 58 2.15 Tensors in a coordinates basis . 61 3 Induced maps, tensor fields and flows 62 3.1 Induced maps: push-forward . 62 3.2 Pull-back . 65 3.3 Vector, covector and tensor fields . 66 3.4 Tensor fields and induced maps . 69 3.5 Induced maps and diffeomorphisms . 72 3.6 Flows . 73 3.7 Lie Derivative of a vector field . 77 3.7.1 Components of the Lie Derivative of a vector field . 78 3.8 Lie Bracket of two vector fields . 81 3.9 Commuting flows . 83 3.10 Lie Derivative of a tensor field . 84 3.11 Example: Active coordinate transformations in GR and sym- metry . 85 4 Differential forms 87 4.1 Cartan wedge product . 88 4.2 Exterior product . 91 4.3 Differential Form Fields . 92 4.4 Exterior derivative . 93 4.5 Pullback of forms . 94 2 4.6 3d vector calculus . 94 4.7 Coordinate free definition of the exterior derivative . 94 4.8 Interior product . 96 4.9 Lie derivative of forms . 96 4.10 Closed and Exact forms . 97 4.11 Physical application: Electromagnetism . 98 4.12 Orientation . 99 4.13 Top forms and volume forms . 100 4.14 Integrating a top form over a chart . 102 4.15 Integrating a top form over an orientable manifold . 104 4.16 Integration of r-forms over oriented r-dimensional submanifolds105 5 Stokes' theorem and cohomology 106 5.1 Cycles and boundaries . 106 5.2 Stokes' theorem . 107 5.3 Example. Gauss's and Stokes' law . 108 5.4 de Rham cohomology . 109 5.5 Example: Cohomology of R . 111 5.6 Example: Cohomology of S1 . 111 5.7 Cohomology and Topology . 113 5.8 Stoke's Theorem and Cohomology . 113 5.9 Poincar´e'sLemma . 115 5.10 Example: Electromagnetism (again!) . 116 5.11 Poincar´eduality . 117 6 Riemannian Geometry I: The metric 118 6.1 The metric . 118 6.2 Metric inner product . 120 6.3 Volume element . 121 6.4 Epsilon symbol . 123 6.5 Hodge Star . 124 6.6 Inner product on r-forms . 125 6.7 Adjoint of d: dy ..........................126 6.8 3d vector calculus again. 128 6.9 The Laplacian . 129 6.10 Ex. Electromagnetism again.... 130 6.11 Hodge theory . 131 6.12 Harmonic representatives for cohomology . 134 3 6.13 Maxwell's Equations on a Compact Riemannian Manifold . 135 7 Riemannian Geometry II: Geometry 137 7.1 Induced metric and volume for a submanifold . 138 7.2 Length of a Curve and the Line Element . 139 7.3 Integration of n-forms over oriented n-dimensional submanifolds140 7.4 Hypersurfaces . 142 7.5 Electric and Magnetic Flux . 145 7.6 Stokes's Theorem in 3D . 145 7.7 Gauss's Theorem . 146 7.8 Connections . 147 7.9 Torsion . 150 7.10 Parallel transport and Geodesics . 152 7.11 Interpretation of Torsion . 153 7.12 Covariant derivative . 155 7.13 Metric connection . 158 7.14 Curvature . 160 7.15 Holonomy . 161 7.16 Non-coordinate basis . 164 7.17 Connections and curvature in non-coordinate bases . 169 7.18 Cartan's structure equations . 171 7.19 Change of Basis and the Local Frame . 174 7.20 Tangent Bundle . 176 4 1 Introduction • A manifold is a space which `locally' looks like Rm (or Cm) • Calculus can be extended from Rm to the manifold. i.e. we have a notion of ‘differentiability'. For a complex manifold, complex analysis can be extended from Cm to the manifold, so we have a notion of `analyticity'. 1.1 Heuristic definition of (real) manifold We essentially require 2 things - an `atlas' which is `smooth' 1. Atlas: m-dim'l manifold M is covered by patches, and there are maps which take patches of M into patches in Rm Each patch of M, Ui, pairs with a map i to give a `chart'. The maps i are invertible and m i : M! R (1) p ! i(p) (2) A set of charts (Ui; i) covering M, so that [iUi = M, is called an `atlas'. −1 0 m The inverse i maps coordinates (x1; : : : ; xm) in Ui 2 R onto Ui 2 M. 5 We explicitly give the maps i by giving m i : M! R (3) p ! x(p) = (x1(p); x2(p); : : : ; xm(p)) (4) Example of an altas: An altas! M = The World (i.e. a 2-sphere), 0 Ui =regions of the world, Ui =pages in altas, i =some appropriate coordinates eg. latitude and longitude. 6 2. Differentiability: transition from one patch to another is smooth. If 2 patches Ui and Uj overlap, so Ui \ Uj 6= 0, then there is a map φji so m 0 m 0 φji : R (Ui ) ! R (Uj) (5) −1 x ! y = φji(x) ≡ j · i (x) (6) −1 for points such that p = i (x) 2 Ui \ Uj. Recall that i is invertible. φji is called the `transition function'. Note that in the diagram, φij should be φji. From the definition we see that φij is invertible since i; j are, and −1 furthermore φij = φji. Since this is a map from Rm ! Rm, our usual notions of differentiability (smoothness) can be used. Explicitly using coordinates where x = (x1; : : : ; xm) and y = (y1; : : : ; ym), we can write the transition function y = φji(x) as ya = ya(x1; : : : ; xm) where a = 1; : : : ; m. 1 Smoothness requirement: Each ya(x1; : : : ; xm) for a = 1; : : : ; m is C in each argument xa. 7 1.2 Example: Stereographic projections of a 2-sphere; the manifold S2 Think of S2 as embedded in R3 as x2 + y2 + z2 = 1. Note that x; y; z provide a unique labelling of points but are not coordinates (there are obviously too many of them!). Consider projecting a point (x; y; z) on the S2 into the plane z = 0 from the North pole. Let the intersection point be (X; Y; 0). Geometrically, x y X = Y = (7) 1 − z 1 − z 2 2 and these define a map N : S ! R . Now X; Y , which are functions of the point p = (x; y; z) look like good coor- dinates. BUT there is a problem at the North pole i.e. p = (0; 0; 1) where obviously something looks suspicious. The pole is mapped to infinity in R2, but which infinity? eg. is it at (0; 1) or (1; 0), or (0; −∞)? Hence the map N is not invertible due to the inclusion of the N-pole. (technically the map fails to be a homeomorphism - see later...) So take 2 charts: project onto plane z = 0 from N-pole for p 2 UN , where UN excludes the N-pole, and project from S-pole onto plane z = 0 for p 2 US, where US excludes the S pole. 8 x y X0 = Y 0 = (8) 1 + z 1 + z 2 2 We should make these charts cover the S and overlap eg. UN = S − 2 fNpoleg, US = S − fSpoleg Differentiability? Y 0 Y 1 + z 1 = X2 + Y 2 = = (9) X0 X 1 − z X02 + Y 02 Transition functions explicitly given as X Y X0 = X0(X; Y ) = Y 0 = Y 0(X; Y ) = (10) X2 + Y 2 X2 + Y 2 define the map 9 This is smooth, C1, in overlap (note that this is true as both poles are ex- cluded, where things are not smooth). Remark: We thought of S2 as a subset of R3 in order to simply label the points. This is a common method of explicitly constructing manifolds, al- though it is not necessary to do this. A very important point is that an atlas provides an intrinsic definition of a manifold { eg. an Atlas is as good as a globe, and describes the same Earth 2-sphere. 1.3 Open sets in Rm in the `usual topology' The canonical example of an open set in Rm is an open ball. m The open ball B(p0) of radius centred on a point p0 in R are the points p in m R whose distance from p0 is less than . If p has coordinates x(p) and p0 has coordinates x0, then the distance between p and p0 is the Euclidean distance 2 2 2 2 jx(p)−x0j (i.e.