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
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                             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
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                             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