Gigli N. Second order analysis on (P2(M), W2). - Providence: American Mathematical Society, 2012. - xii, 154 p. - (Memoirs of the American Mathematical Society; vol.216, N 1018). - Bibliogr.: p.153-154. - ISSN 0065-9266; ISBN 978-0-8218-5309-2  Место хранения:   013 | Институт математики СО РАН | Новосибирск | Библиотека

Оглавление / Contents

Introduction .................................................. vii Chapter 1 Preliminaries and notation ........................... 1 1.1 Riemannian manifolds ....................................... 2 1.2 The distance W2 ............................................ 5 1.3 Kantorovich's dual problem ................................. 9 1.4 First order differentiable structure ...................... 15 Chapter 2 Regular curves ...................................... 21 2.1 Cauchy Lipschitz theory on Riemannian manifolds ........... 21 2.2 Definition and first properties of regular curves ......... 24 2.3 On the regularity of geodesies ............................ 29 Chapter 3 Absolutely continuous vector fields ................. 39 3.1 Definition and first properties ........................... 39 3.2 Approximation of absolutely continuous vector fields ...... 44 Chapter 4 Parallel transport .................................. 49 4.1 The case of an embedded Riemannian manifold ............... 49 4.2 Parallel transport along regular curves ................... 52 4.3 Forward and backward parallel transport ................... 57 4.4 On the question of stability and the continuity of μ → Pμ .................................................... 60 Chapter 5 Covariant derivative ................................ 65 5.1 Levi-Civita connection .................................... 65 5.2 The tensor Nμ ............................................ 72 5.3 Calculus of derivatives ................................... 77 5.4 Smoothness of time dependent operators .................... 86 Chapter 6 Curvature ........................................... 93 6.1 The curvature tensor ...................................... 93 6.2 Related notions of curvature .............................. 98 Chapter 7 Differentiability of the exponential map ........... 101 7.1 Introduction to the problem .............................. 101 7.2 Rigorous result .......................................... 103 7.3 A pointwise result ....................................... 107 Chapter 8 Jacobi fields ...................................... 113 8.1 The Jacobi equation ...................................... 113 8.2 Solutions of the Jacobi equation ......................... 115 8.3 Points before the first conjugate ........................ 119 Appendix A Density of regular curves ......................... 121 Appendix В C1 curves ......................................... 131 Appendix C On the definition of exponential map .............. 139 Appendix D A weak notion of absolute continuity of vector fields ............................................ 147 Bibliography .................................................. 153