135x Filetype PDF File size 0.94 MB Source: webspace.science.uu.nl
Mastermath course Differential Geometry 2015/2016 Marius Crainic 1Department of Mathematics, Utrecht University, 3508 TA Utrecht, The Netherlands. E-mail address: m.crainic@uu.nl. Received by the editors September, 2015. c 0000 American Mathematical Society 1 Contents Mastermath course Differential Geometry 2015/2016 1 Chapter1. Vector bundles and connections 9 1.1. Vector bundles 9 1.1.1. The definition/terminology 9 1.1.2. Morphisms 10 1.1.3. Trivial vector bundles; trivializations 10 1.1.4. Sections 10 1.1.5. Frames 11 1.1.6. Remark on the construction of vector bundles 11 1.1.7. Operations with vector bundles 13 1.1.8. Differential forms with coefficients in vector bundles 18 1.2. Connections on vector bundles 19 1.2.1. The definition 19 1.2.2. Locality; connection matrices 21 1.2.3. More than locality: derivatives of paths 21 1.2.4. Parallel transport 23 1.2.5. Connections as DeRham-like operators 24 1.3. Curvature 25 1.3.1. The definition 25 2 2 1.3.2. The curvature as failure of ” d = d ” 27 dsdt dtds 1.3.3. The curvature as failure of d2 = 0 28 ∇ 1.3.4. More on connection and curvature matrices; the first Chern class 29 1.4. Connections compatible with a metric 35 1.4.1. Metrics on vector bundles 35 1.4.2. Connections compatible with a metric 37 1.4.3. Riemannian manifolds I: The Levi-Civita connection 38 1.4.4. Riemannian manifolds II: geodesics and the exponential map 39 1.4.5. Riemannian manifolds III: application to tubular neighborhoods 42 Chapter2. Principal bundles 47 2.1. Digression: Lie groups and actions 47 2.1.1. The basic definitions 47 2.1.2. The Lie algebra 50 2.1.3. The exponential map 54 2.1.4. Closed subgroups of GL 55 n 2.1.5. Infinitesimal actions 57 2.1.6. Free and proper actions 58 2.2. Principal bundles 61 3 4 M. CRAINIC, DG-2015 2.2.1. The definition/terminology 61 2.2.2. Remarks on the definition 62 2.2.3. From vector bundles to principal bundles (the frame bundle) 66 2.2.4. From principal bundles to vector bundles 68 2.2.5. Sections/forms for vector bundles associated to principal ones 70 2.3. Connections on principal bundles 73 2.3.1. Connections as horizontal distributions: 73 2.3.2. Connection forms 74 2.3.3. Parallel transport 75 2.3.4. Curvature 77 2.3.5. Principal bundle connections ←→ vector bundles ones 78 2.3.6. Reduction of the structure group 80 Chapter3. G-structures by examples 83 3.1. Geometric structures on vector spaces 83 3.1.1. Linear G-structures 83 3.1.2. Example: Inner products 85 3.1.3. Example: Orientations 86 3.1.4. Example: Volume elements 87 3.1.5. Example: p-directions 88 3.1.6. Example: Integral affine structures 89 3.1.7. Example: Complex structures 89 3.1.8. Example: Symplectic forms 91 3.1.9. Hermitian structures 94 3.1.10. Afew more remarks on linear G-structures 96 3.2. Geometric structures on manifolds 98 3.2.1. G-structures on manifolds 98 3.2.2. G={e}: frames and coframes 99 3.2.3. Example: Orientations 100 3.2.4. Example: Volume elements 103 3.2.5. Example: foliations 105 3.2.6. Example: Integral affine structures 107 3.2.7. Example: Complex structures 108 3.2.8. Example: Symplectic forms 111 3.2.9. Example: Affine structures 114 Chapter4. G-structures and connections 117 4.1. Compatible connections 117 4.1.1. Connections compatible with a G-structure 117 4.1.2. The infinitesimal automorphism bundle 120 4.1.3. The compatibility tensor 121 4.1.4. The space of compatible connections 123 4.2. Torsion and curvature (and their relevance to integrability) 124 4.2.1. Some linear algebra that comes with G-structures 124 4.2.2. The torsion of compatible connections; the intrinsic torsion 126 4.2.3. The curvature of compatible connections; the intrinsic curvature 128 4.2.4. Example: Riemannian metrics 131 4.2.5. Example of the intrinsic torsion: symplectic forms 132 4.2.6. Example of the intrinsic torsion: complex structures 133
no reviews yet
Please Login to review.