## Global Analysis in Mathematical Physics: Geometric and Stochastic MethodsThe 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. |

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### Contents

IV | 3 |

V | 5 |

VI | 7 |

VII | 10 |

IX | 12 |

X | 14 |

XI | 17 |

XII | 19 |

L | 106 |

LI | 107 |

LII | 109 |

LIV | 114 |

LV | 115 |

LVI | 125 |

LVII | 131 |

LVIII | 133 |

XIII | 20 |

XIV | 22 |

XVI | 23 |

XVII | 24 |

XVIII | 26 |

XIX | 28 |

XX | 29 |

XXI | 31 |

XXII | 32 |

XXIII | 39 |

XXIV | 40 |

XXV | 42 |

XXVI | 45 |

XXVII | 47 |

XXVIII | 49 |

XXIX | 50 |

XXX | 53 |

XXXI | 54 |

XXXII | 56 |

XXXIII | 57 |

XXXIV | 58 |

XXXV | 59 |

XXXVI | 67 |

XXXVII | 76 |

XXXIX | 78 |

XL | 80 |

XLI | 83 |

XLII | 87 |

XLIV | 92 |

XLV | 95 |

XLVI | 96 |

XLVII | 97 |

XLVIII | 98 |

XLIX | 101 |

### Other editions - View all

Global Analysis in Mathematical Physics: Geometric and Stochastic Methods Yuri Gliklikh Limited preview - 2012 |

Global Analysis in Mathematical Physics: Geometric and Stochastic Methods Yuri Gliklikh No preview available - 2012 |

### Common terms and phrases

analog Appendix apply Assume Aſt boundary bounded bundle called Chap chart clear closed compact complete connection Consider constant constraint construction continuous coordinates covariant curve defined definition Denote derivative diffeomorphisms differential equations diffusion easy energy example exists fact fibers fixed flow follows force field formula function geodesic geometry given grad ideal incompressible fluid independent initial condition inner product integral interval introduce Itô equation Lemma linear manifold means measure mechanical system method motion Namely natural neighborhood Newton equation Note obtain operator parallel translation particular probability problem proof properties prove Recall Remark respect Riemannian manifold Riemannian metric right-invariant satisfies Sect similar smooth solution standard stochastic differential stochastic differential equations stochastic process tangent space Theorem theory trajectory turns uniformly unique values vector field weak Wiener process

### Popular passages

Page x - In conclusion, the author would like to express his deep gratitude to Candidates of Physicomathematical Sciences, AI Kuz'min and LI Dorman for their valuable advice.

Page viii - Newton equation is introduced by means of the covariant derivative with respect to the Levi-Civita connection of the Riemannian metric, giving rise to the kinetic energy on the configuration space.