Lecture notes on sub-Riemannian geometry

A Brief Introduction*......Page 9
   Analysis of hypoelliptic operators......Page 13
   Univalent Function Theory......Page 14
   Falling cats......Page 15
   From mechanics: parking cars, rolling balls, moving robots, and satellites......Page 16
   Vision......Page 17
   Quantum mechanical systems......Page 18
   Even more examples......Page 19
 An isoperimetric problem on the plane......Page 21
 The contact-geometry formulation of the problem......Page 22
 The Heisenberg group......Page 25
  The 3D nilpotent non-Abelian matrix group......Page 26
 The subRiemannian Heisenberg group......Page 27
  The geodesics and the spheres in the Heisenberg group......Page 28
  Dilations on the Heisenberg group......Page 31
   The dimension of the Heisenberg group......Page 33
   More pictures......Page 34
 Exercises......Page 35
  Metric Spaces......Page 37
  Length of curves in metric spaces......Page 38
  Length Space, Intrinsic Metrics, and Geodesic Spaces......Page 40
  Isometries and Lipschitz maps......Page 41
  Hausdorff Measures and Dimension......Page 42
  Vector fields and Lie brackets......Page 45
  Vector bundles......Page 47
  Riemannian and Finsler geometry......Page 48
  Bracket-generating distributions......Page 50
  SubFinsler structures......Page 51
  The generalization of Control Theory......Page 52
  The general definition......Page 53
 Chow's Theorem and existence of geodesics......Page 54
  Reachable sets of bracket-generating distributions......Page 55
  The metric version of Chow's theorem......Page 56
  Comparison of length structures......Page 57
 Equiregular Distributions......Page 58
  Ball-Box Theorem......Page 60
  Dimensions of CC spaces......Page 62
 Exercises......Page 63
 A review of Lie groups and their Lie algebras......Page 67
  Exponential map......Page 69
  Nilpotent Lie groups and nilpotent Lie algebras......Page 71
  Examples......Page 72
  Matrix groups......Page 74
  Adjoint operators......Page 75
  Simply connected nilpotent Lie groups......Page 76
  Homogeneous manifolds......Page 82
 Stratified Lie algebras......Page 83
  Some elementary facts about stratifications......Page 84
  Dilation Structures......Page 86
 Carnot groups......Page 87
  A good basis for a Carnot group......Page 90
  Local-to-global using dilations, and canonical coordinates......Page 91
   The brackets as products of exponentials......Page 93
  A proof of the Ball-Box Theorem for Carnot groups......Page 97
  Haar, Hausdorff and Lebesgue measures......Page 99
 Exercises......Page 100
  A topology on the space of metric spaces......Page 105
  Asymptotic cones and tangent spaces......Page 106
  More general limits of Riemannian manifolds......Page 107
  Preparatory example: The Riemannian Heisenberg group......Page 108
  Toward the general setting: Gronwall Lemma......Page 109
  Asymptotic cones of Riemannian stratified groups......Page 111
   Quantitative Chow's theorem......Page 112
   Proof of Theorem 4.3.1......Page 114
  Nilpotization......Page 116
  Mitchell's Theorem on tangent cones......Page 117
 A metric characterization of Carnot groups......Page 118
  Proof of the characterization......Page 119
 Heisenberg group as boundary......Page 121
  Margulis-Mostow's theorem......Page 123
   Preliminaries to the proof of Pansu's theorem......Page 124
   Proof of Pansu's theorem......Page 125
  Applications to non-embeddability......Page 129
   Comments regarding minimal surfaces and isoperimetric solutions......Page 131
  A review of divergence and distributions......Page 132
  Caccioppoli sets: sets of locally finite perimeter......Page 134
  Notions of rectificability......Page 135
  Results on geodesics......Page 137
  Results on metric spheres......Page 138
  Results on the isoperimetric problem......Page 139
  Results on minimal surfaces......Page 140
  More results on regularity......Page 142
 Translations and flows......Page 143
  X-derivative of nice functions and domains......Page 144
 Exercises......Page 145
 Elements of Geometric Group Theory......Page 147
  Polynomial growth and virtual nilpotency......Page 150
 Asymptotic cone......Page 151
 The Malcev closure......Page 152
 The limit CC metric......Page 153
Equivalent Definitions of Length......Page 155
  The variation of length......Page 159
  The area functional and its variation......Page 160
  The conclusion......Page 162
 A special sub-Finsler geometry on nilpotent groups......Page 165
  The ``development'' of a curve......Page 166
  Horizontal lines as geodesics......Page 167
  Lifts of curves......Page 169
  Curves in free nilpotent Lie groups......Page 172
  Open questions......Page 174
Bibliography......Page 175
Index......Page 177