Lectures on nonsmooth differential geometry

Preface......Page 6
Contents......Page 9
About the Authors......Page 11
  1.1.1 Borel Probability Measures......Page 12
  1.1.2 The Space L0(m)......Page 21
  1.1.3 Pushforward of Measures......Page 24
 1.2 Spaces of Curves......Page 25
 1.3 Bochner Integral......Page 31
 Bibliographical Remarks......Page 42
  2.1.1 Test Plans......Page 43
  2.1.2 Definition of Sobolev Space......Page 45
  2.1.3 Calculus Rules......Page 53
  2.1.4 Local Sobolev Space......Page 56
  2.1.5 Consistency with the Classical Sobolev Space on Rn......Page 59
  2.2.1 Approach à la Cheeger......Page 62
  2.2.2 Approach à la Shanmugalingam......Page 68
 Bibliographical Remarks......Page 74
  3.1.1 L2-Normed Linfty-Modules......Page 76
  3.1.2 L0-Normed L0-Modules......Page 81
  3.2.1 Dual Normed Module......Page 85
  3.2.2 Hilbert Modules and Tensor Products......Page 91
  3.2.3 Pullback of Normed Modules......Page 97
 Bibliographical Remarks......Page 105
  4.1.1 Definition and Basic Properties......Page 106
  4.1.2 Calculus Rules and Their Consequences......Page 110
  4.2.1 Definition and Basic Properties......Page 115
  4.2.2 Divergence Operator and Gradients......Page 116
 4.3 Infinitesimal Hilbertianity......Page 121
 4.4 Maps of Bounded Deformation......Page 125
 Bibliographical Remarks......Page 131
  5.1.1 Set-Up of the Theory......Page 132
  5.1.2 Existence and Uniqueness of the Gradient Flow......Page 137
  5.2.1 Laplace Operator......Page 142
  5.2.2 Heat Flow and Its Properties......Page 145
 Bibliographical Remarks......Page 151
  6.1.1 Definition of RCD Space......Page 152
  6.1.2 Heat Flow on RCD Spaces......Page 154
  6.1.3 Test Functions......Page 156
  6.2.1 Definition and Basic Properties......Page 158
  6.2.2 Measure-Valued Laplacian......Page 161
  6.2.3 Presence of Many W22-Functions......Page 166
  6.2.4 Calculus Rules......Page 171
 6.3 Covariant Derivative......Page 174
  6.4.1 Sobolev Differential Forms......Page 177
  6.4.2 de Rham Cohomology and Hodge Theorem......Page 178
 6.5 Ricci Curvature Operator......Page 186
 Bibliographical Remarks......Page 187
A Functional Analytic Tools......Page 189
B Solutions to the Exercises......Page 191
References......Page 205
Notation......Page 207
Index......Page 211