Global Analysis in Mathematical Physics, Softcover reprint of the original 1st ed. 1997 Geometric and Stochastic Methods Applied Mathematical Sciences Series, Vol. 122

The first edition of this book entitled Analysis on Riemannian Manifolds and Some Problems of Mathematical Physics was published by Voronezh Univer­ sity 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. I am grateful to Viktor L. Ginzburg for his hard work on the transla­ tion and for writing Appendix F, and to Tomasz Zastawniak for his numerous suggestions. My special thanks go to the referee for his valuable remarks on the theory of stochastic processes. Finally, I would like to acknowledge the support of the AMS fSU Aid Fund and the International Science Foundation (Grant NZBOOO), which made possible my work on some of the new results included in the English edition of the book. Voronezh, Russia Yuri Gliklikh September, 1995 Preface to the Russian Edition The present book is apparently the first in monographic literature in which a common treatment is given to three areas of global analysis previously consid­ ered quite distant from each other, namely, differential geometry and classical mechanics, stochastic differential geometry and statistical and quantum me­ chanics, and infinite-dimensional differential geometry of groups of diffeomor­ phisms and hydrodynamics. The unification of these topics under the cover of one book appears, however, quite natural, since the exposition is based on a geometrically invariant form of the Newton equation and its analogs taken as a fundamental law of motion.

I. Finite-Dimensional Differential Geometry and Mechanics.- 1 Some Geometric Constructions in Calculus on Manifolds.- 2 Geometric Formalism of Newtonian Mechanics.- 3 Accessible Points of Mechanical Systems.- II. Stochastic Differential Geometry and its Applications to Physics.- 4 Stochastic Differential Equations on Riemannian Manifolds.- 5 The Langevin Equation.- 6 Mean Derivatives, Nelson’s Stochastic Mechanics, and Quantization.- III. Infinite-Dimensional Differential Geometry and Hydrodynamics.- 7 Geometry of Manifolds of Diffeomorphisms.- 8 Lagrangian Formalism of Hydrodynamics of an Ideal Incompressible Fluid.- 9 Hydrodynamics of a Viscous Incompressible Fluid and Stochastic Differential Geometry of Groups of Diffeomorphisms.- Appendices.- A. Introduction to the Theory of Connections.- Connections on Principal Bundles.- Connections on the Tangent Bundle.- Covariant Derivatives.- Connection Coefficients 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 Itô Group and the Principal Itô Bundle.- E. Sobolev Spaces.- F. Accessible Points and Closed Trajectories 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.

This book gives a common treatment to three areas of application of global analysis in mathematical physics previously considered quite distant of each other, namely, differential geometry applied to classical mechanics, stochastic differential geometry used in quantum and statistical mechanics, and infinite-dimensional differential geometry fundamental for hydrodynamics. The unification of these topics is made possible by considering Newton equation or its natural generalizations and analogues as a fundamental equation of motion.