The first edition of this book entitled Analysis on Riemannian Manifolds and Some Problems of Mathematical Physics was published by Voronezh University Press in 1989. For its English edition, the book has been substantially revised and expanded. In particular, new material has been added to Sections 19 and 20.

Preface to the English Edition

Preface to the Russian Edition

Part I．Finite-Dimensional Difrerential Geometry and Mechanics

 Chapter 1

Some Geometric C0nstructions in Calculus on Manifolds 

1．Complete Riemannian Metrics and the Completeness of Vector Fields

1．A A Necessary and Sufficient Condition for the Completeness of a Vector Field

1．B A Way to Construct Complete Riemannian Metrics

2． Riemannian Manifolds Possessing a Uniform Riemannian Atlas

3． Integral Operators with Parallel Translation

3．A The Operator S

3．B The 0perator

3．C Integral 0perators

 Chapter 2

Geometric Formalism of Newtonian Mechanics

4．Geometric Mechanics：Introduction and Review of Standard Examples

4．A Basic Notions

4．B Some Special Classes of Force Fields

4．C Mechanical Systems on Groups

5．Geometric Mechanics with Linear Constraints

5．A Linear Mechanical Constraints

5．B Reduced Connections

5．C Length Minimizing and Least—Constrained Nonholonomic Geodesics

6．Mechanical Systems with Discontinuous Forces and Systems with C0ntroh Difrerential Inclusions

7．Integral Equations of Geometric Mechanics： The Velocity Hodograph

7．A General Constructions

7．B Integral Formalism of Geometric Mechanics with Constraints

8．Mechanical Interpretation of Parallel Translation and Systems with Delayed Control Force

 Chapter 3

Accessible Points of Mechanical Systems

9． Examples of Points that Cannot Be Connected by a Trajectory

10．The Main Result on Accessible Points

11．Generalizations to Systems with Constraints

Part Ⅱ．Stochastic Differential Geometry and its Applications to Physics

 Chapter 4

Stochastic Differential Equations on Riemannian Manifolds一

12．Review of the Theory of Stochastic Equations and Integrals on Finite Dimensional Linear Spaces

12．A Wiener Processes

12．B The Ite Integral

12．C The Backward Integral and the Stratonovich Integral

12．D The Ite and Stratonovich Stochastic Differential Equations

12．E Solutions of SDEs

12．F Approximation by Solutions of Ordinary Differential Equations

12．G A Relationship Between SDEs and PDEs

13．Stochastic Differential Equations on Manifolds

14．Stochastic Parallel Translation and the Integral Formalism for the It Equations

15．Wiener Processes on Riemannian Manifolds and Related Stochastic Difierential Equations

15．A Wiener Processes on Riemannian Manifolds

15．B Stochastic Equations

15．C Equations with Identity as the Diffusion Coefficient

16．Stochastic Differential Equations with Constraints

 Chapter 5

The Langevin Equation

17．The Langevin Equation ofGeometric Mechanics

18．Strong Solutions of the Langevin Equation， Ornstein——Uhlenbeck Processes

 Chapter 6

Mean Derivatives，Nelson’s Stochastic Mechanics，and Quantization

19．More on Stochastic Equations and Stochastic Mechanics in ■

19．A Preliminaries

19．B Forward Mean Derivatives

19．C Backward Mean Derivatives and Backward Equations

19．D Symmetric and Antisymmetric Derivatives

19．E The Derivatives of a Vector Field Along ■ and the Acceleration of ■

19．F Stochastic Mechanics

20．Mean Derivatives and Stochastic Mechanics on Riemannian Manifolds

20．A Mean Derivatives on ManifoIds and Related Equations

20．B Geometric Stochastic Mechanics

20．C The Existence of Solutions in Stochastic Mechanics

21．Relativistic Stochastic Mechanics

Part Ⅲ．Inflnite-Dimensional Difrerential Geometry and Hydrodynamics

 Chapter 7

Geometry 0f Manifolds 0f Diffomorphisms

22．Manifolds of Mappings and Groups of Diffeomorphisms

22．A Manifolds of Mappings

22．B The Group of Hs-Diomorphisms

22．C Diffeomorphisms of a Manifold with Boundary 

22．D Some Smooth Operators and Vector Bundles over ■

23．Wleak Riemannian Metrics and Connections on Manifolds of Difreomorphisms 

23．A The Case of a Closed Manifold

23．B The Case of a Manifold with Boundary

23．C The Strong Riemannian Metric

24．Lagrangian Formalism of Hydrodynamics of an Ideal Barotropic Fluid

24．A Diffhse Matter 

24．B A Barotropic Fluid

 Chapter 8

Lagrangian Formalism of Hydrodynamics of an Ideal

Incompressible Fluid

25．Geometry of the Manifold of Volume-Preserving Diffeomorphisms and LHSs of an Ideal Incompressible Fluid

25．A V．0lume-Preserving Diffeomorphisms of a Closed Manifold

25．B V．0lume-Preserving Diffeomorphisms of a Manifold with Boundary

25．C LHS’S of an Ideal Incompressible Fluid

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

27．The Regularity Theorem and a Review of Resuits on the Existence of Solutions

 Chapter 9

Hydrodynamics of a Viscous Incompressible Fluid and Stochastic Difrerential Geometry 0f Groups 0f Difieomorphisms

28．Stochastic Difierential Geometry on the Groups of Diffeomorphisms of the n-Dimensional Torus

29．A Viscous Incompressible Fluid

Appendices

 A．Introduction to the Theory of Connections

Connections on Principal Bundles

Connections on the Tangent Bundle

Covariant Derivatives

Connection Coefflcients and Christoffel Symbols

Second-Order Differential Equations and the Spray

The Exponential Map and Normal Charts

 B．Introduction to the Theory of Set—Valued Maps

 C．Basic Definitions of Probability Theory and

the Theory of Stochastic Processes

Stochastic Processes and Cylinder Sets

The Conditional Expectation

Markovian Processes

Martingales and Semimartingales

 D．The Ite Group and the Principal It5 Bundle

 E．Sobolev Spaces

 F．Accessible Points and Closed najectories of

Mechanical Systems(by Viktor L．Ginzburg)

Growth of the Force Field and Accessible Points

Accessible Points in Systems with Constraints

Closed Trajectories of Mechanical Systems

References

Index