Part Ⅰ．Finite-Dimensional Differential Geometry and Mechanics 3
Chapter 1 Some Geometric Constructions in Calculus on Manifolds 3
1．Complete Riemannian Metrics and the Completeness of Vector Fields 3
1.A A Necessary and Sufficient Condition for the Completeness of a Vector Field 3
1.B A Way to Construct Complete Riemannian Metrics 5
2．Riemannian Manifolds Possessing a Uniform Riemannian Atlas 7
3．Integral Operators with Parallel Translation 10
3.A The Operator S 10
3.B The Operator Γ 12
3.C Integral Operators 14
Chapter 2 Geometric Formalism of Newtonian Mechanics 17
4．Geometric Mechanics:Introduction and Review of Standard Examples 17
4.A Basic Notions 17
4.B Some Special Classes of Force Fields 19
4.C Mechanical Systems on Groups 20
5．Geometric Mechanics with Linear Constraints 22
5.A Linear Mechanical Constraints 22
5.B Reduced Connections 23
5.C Length Minimizing and Least-Constrained Nonholonomic Geodesics 24
6．Mechanical Systems with Discontinuous Forces and Systems with Control:Differential Inclusions 26
7．Integral Equations of Geometric Mechanics:The Velocity Hodograph 28
7.A General Constructions 29
7.B Integral Formalism of Geometric Mechanics with Constraints 31
8．Mechanical Interpretation of Parallel Translation and Systems with Delayed Control Force 32
Chapter 3 Accessible Points of Mechanical Systems 39
9．Examples of Points that Cannot Be Connected by a Trajectory 40
10．The Main Result on Accessible Points 41
11．Generalizations to Systems with Constraints 45
Part Ⅱ．Stochastic Differential Geometry and its Applications to Physics 49
Chapter 4 Stochastic Differential Equations on Riemannian Manifolds 49
12．Review of the Theory of Stochastic Equations and Integrals on Finite-Dimensional Linear Spaces 49
12.A Wiener Processes 49
12.B The It？ Integral 50
12.C The Backward Integral and the Stratonovich Integral 53
12.D The It？ and Stratonovich Stochastic Differential Equations 54
12.E Solutions of SDEs 56
12.F Approximation by Solutions of Ordinary Differential Equations 57
12.G A Relationship Between SDEs and PDEs 58
13．Stochastic Differential Equations on Manifolds 59
14．Stochastic Parallel Translation and the Integral Formalism for the It？ Equations 67
15．Wiener Processes on Riemannian Manifolds and Related Stochastic Differential Equations 76
15.A Wiener Processes on Riemannian Manifolds 76
15.B Stochastic Equations 78
15.C Equations with Identity as the Diffusion Coefficient 80
16．Stochastic Differential Equations with Constraints 83
Chapter 5 The Langevin Equation 87
17．The Langevin Equation of Geometric Mechanics 87
18．Strong Solutions of the Langevin Equation,Ornstein-Uhlenbeck Processes 91
Chapter 6 Mean Derivatives,Nelson's Stochastic Mechanics,and Quantization 95
19．More on Stochastic Equations and Stochastic Mechanics in Rn 96
19.A Preliminaries 96
19.B Forward Mean Derivatives 97
19.C Backward Mean Derivatives and Backward Equations 98
19.D Symmetric and Antisymmetric Derivatives 101
19.E The Derivatives of a Vector Field Along ξ(t)and the Acceleration of ξ(t) 106
19.F Stochastic Mechanics 107
20．Mean Derivatives and Stochastic Mechanics on Riemannian Manifolds 109
20.A Mean Derivatives on Manifolds and Related Equations 109
20.B Geometric Stochastic Mechanics 114
20.C The Existence of Solutions in Stochastic Mechanics 115
21．Relativistic Stochastic Mechanics 125
Part Ⅲ．Infinite-Dimensional Differential Geometry and Hydrodynamics 133
Chapter 7 Geometry of Manifolds of Diffeomorphisms 133
22．Manifolds of Mappings and Groups of Diffeomorphisms 133
22.A Manifolds of Mappings 133
22.B The Group of Hs-Diffeomorphisms 134
22.C Diffeomorphisms of a Manifold with Boundary 136
22.D Some Smooth Operators and Vector Bundles over Ds(M) 137
23．Weak Riemannian Metrics and Connections on Manifolds of Diffeomorphisms 139
23.A The Case of a Closed Manifold 139
23.B The Case of a Manifold with Boundary 141
23.C The Strong Riemannian Metric 141
24．Lagrangian Formalism of Hydrodynamics of an Ideal Barotropic Fluid 142
24.A Diffuse Matter 142
24.B A Barotropic Fluid 143
Chapter 8 Lagrangian Formalism of Hydrodynamics of an Ideal Incompressible Fluid 147
25．Geometry of the Manifold of Volume-Preserving Diffeomorphisms and LHSs of an Ideal Incompressible Fluid 147
25.A Volume-Preserving Diffeomorphisms of a Closed Manifold 148
25.B Volume-Preserving Diffeomorphisms of a Manifold with Boundary 151
25.C LHS's of an Ideal Incompressible Fluid 152
26．The Flow of an Ideal Incompressible Fluid on a Manifold with Boundary as an LHS with an Infinite-Dimensional Constraint on the Group of Diffeomorphisms of a Closed Manifold 156
27．The Regularity Theorem and a Review of Results on the Existence of Solutions 164
Chapter 9 Hydrodynamics of a Viscous Incompressible Fluid and Stochastic Differential Geometry of Groups of Diffeomorphisms 171
28．Stochastic Differential Geometry on the Groups of Diffeomorphisms of the n-Dimensional Torus 172
29．A Viscous Incompressible Fluid 175
Appendices 179
A．Introduction to the Theory of Connections 179
Connections on Principal Bundles 179
Connections on the Tangent Bundle 180
Covariant Derivatives 181
Connection Coefficients and Christoffel Symbols 183
Second-Order Differential Equations and the Spray 185
The Exponential Map and Normal Charts 186
B．Introduction to the Theory of Set-Valued Maps 186
C．Basic Definitions of Probability Theory and the Theory of Stochastic Processes 188
Stochastic Processes and Cylinder Sets 188
The Conditional Expectation 188
Markovian Processes 189
Martingales and Semimartingales 190
D．The It？ Group and the Principal It？ Bundle 190
E．Sobolev Spaces 191
F．Accessible Points and Closed Trajectories of Mechanical Systems(by Viktor L.Ginzburg) 192
Growth of the Force Field and Accessible Points 193
Accessible Points in Systems with Constraints 197
Closed Trajectories of Mechanical Systems 198
References 203
Index 211