Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics

Подождите немного. Документ загружается.

Global and Stochastic Analysis with Applications

to Mathematical Physics

Theoretical and Mathematical Physics

The series founded in 1975 and formerly (until 2005) entitled Texts and Monographs

in Physics (TMP) publishes high-level monographs in theoretical and mathematical

physics. The change of title to Theoretical and Mathematical Physics (TMP) signals

that the series is a suitable publication platform for both the mathematical and the the-

oretical physicist. The wider scope of the series is reﬂected by the composition of the

editorial board, comprising both physicists and mathematicians.

The books, written in a didactic style and containing a certain amount of elementary

background material, bridge the gap between advanced textbooks and research mono-

graphs. They can thus serve as basis for advanced studies, not only for lectures and

seminars at graduate level, but also for scientists entering a ﬁeld of research.

Editorial Board

W. Beiglboeck, Institute of Applied Mathematics, University of Heidelberg, Heidelberg,

Germany

P. Chrusciel, Gravitational Physics, University of Vienna, Vienna, Austria

J.-P. Eckmann, Université de Genève, Département de Physique Théorique, Geneva,

Switzerland

H. Grosse, Institute of Theoretical Physics, University of Vienna, Vienna, Austria

A. Kupiainen, Department of Mathematics, University of Helsinki, Helsinki, Finland

M. Loss, School of Mathematics, Georgia Institute of Technology, Atlanta, USA

H. Löwen, Institute of Theoretical Physics, Heinrich-Heine-University of Duesseldorf,

Duesseldorf, Germany

N. Nekrasov, IHÉS, Bures-sur-Yvette, France

M. Salmhofer, Institute of Theoretical Physics, University of Heidelberg, Heidelberg,

Germany

S. Smirnov, Mathematics Section, University of Geneva, Geneva, Switzerland

L. Takhtajan, Department of Mathematics, Stony Brook University, Stony Brook, USA

J. Yngvason, Institute of Theoretical Physics, University of Vienna, Vienna, Austria

For other titles published in this series, go to

www.springer.com/series/720

Yuri E. Gliklikh

Global and

Stochastic Analysis

with Applications

to Mathematical

Physics

Prof. Yuri E. Gliklikh

Voronezh State University

Mathematics Department

Universitetskaya pl. 1

394006 Voronezh

Russia

yeg@math.vsu.ru

ISSN 1864-5879

ISBN 978-0-85729-162-2

e-ISSN 1864-5887

e-ISBN 978-0-85729-163-9

DOI 10.1007/978-0-85729-163-9

Springer London Dordrecht Heidelberg New York

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

Mathematics Subject Classiﬁcation (2010): 58-02, 60-02, 58D30, 58J65, 60H30, 70G45, 76M35, 81P20,

82C31

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as per-

mitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,

stored or transmitted, in any form or by any means, with the prior permission in writing of the publish-

ers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the

the publishers.

The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a

speciﬁc statement, that such names are exempt from the relevant laws and regulations and therefore free

for general use.

The publisher makes no representation, express or implied, with regard to the accuracy of the information

contained in this book and cannot accept any legal responsibility or liability for any errors or omissions

that may be made.

Cover design: eStudio Calamar S.L.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

To the memory of my parents and my wife

Olga

Preface

Since the publication of my previous books [99, 106], many papers and mono-

graphs have appeared which develop similar themes. So, now is a good time

for an “intermediate ﬁnish”, a new book containing results obtained up to

the present.

The themes of the book cross the borders of several mathematical subjects

including the geometry of manifolds, stochastic analysis, set-valued analysis

and some chapters of mathematical physics. Thus, an important feature of the

book is that it includes a signiﬁcant amount of preliminary material. However,

my original intention to make the book completely self-contained yielded such

an incredible volume of text that I had to reduce the preliminaries and assume

the reader to be at least a little familiar with the aforementioned branches

of mathematics. Nevertheless, I hope that the remaining preliminaries will

be useful for an expert in one of the subjects who wishes to understand the

others and will also familiarize non-experts with the general themes of this

work.

I want to express my thanks to a lot of people who aided me in my research.

First of all, I am obliged to my advisor Yu.G. Borisovich who introduced me

to mathematics. In particular, I owe my interest in Global Analysis and its

applications to Mathematical Physics to him. Another strong inﬂuence (this

time in Stochastic Analysis) was exerted on my research by K.D. Elworthy. I

am indebted to him for long and extremely useful discussions of the problems

that interested me, as well as for repeated invitations to Warwick University,

allowing me to ventilate my ideas with many mathematicians from all over

the world. In addition, I should specially mention Yu.L. Daletski˘ıwhowas

also a great help.

I express my gratitude to my colleagues and co-authors: Ya.I. Belopolskaya,

B.D. Gelman, I.V. Fedorenko and T. Zastawniak and my former students and

co-authors: S.V. Azarina, Yu.S. Baranov, L.A. Morozova, A.V. Obukhovskii,

P.S. Ratiner and P.S. Zykov for their interest in our joint work. I am grate-

ful to Z. Brzezniak, V.Ya. Gershkovich, A.I. Shnirelman, A. Truman, A.M.

Vershik and many others for very useful discussions.

vii

viii Preface

My research, the results of which are included in this book, was supported

in part by INTAS Grants 94-00378 and 99-00559, RFBR Grants 96-01-00444,

99-01-00819, 03-01-00112, 04-01-00081, 07-01-00137 and 08-01-00155 as well

as by Grants UR.04.01.003 and UR.04.01.008 of the program “Universities

of Russia” and by Grant VZ-010-0 of the Russian Ministry of Education and

CRDF. I am grateful to all these foundations.

Yuri E. Gliklikh

September, 2009

Contents

Part I Global Analysis

1 Manifolds and Related Objects ........................... 3

1.1 Manifolds,Vectorsand Covectors. AGlossary.............. 3

1.2 LieGroups and LieAlgebras............................. 11

1.3 Fiber Bundles . ......................................... 14

1.4 Riemannianand Semi-Riemannian Metrics ................ 18

1.5 Tensors ............................................... 21

1.6 DiﬀerentialForms and Polyvectors........................ 25

1.7 TheLie Derivative...................................... 32

2 Connections .............................................. 35

2.1 The Structure of a Tangent Bundle to a Vector Bundle . . . . . 35

2.2 Connections on Vector Bundles . .......................... 40

2.3 Connectionson Manifolds ............................... 50

2.4 Geodesics ............................................. 53

2.5 Curvatureand TorsionTensors........................... 55

2.6 Riemannian Connections. The Levi-Civit´a Connection . . . . . . 56

2.7 Connections on Principal Bundles . . ...................... 60

2.8 A Connection on the Total Space of a Vector Bundle ........ 64

2.9 SecondOrder TangentVectors and Connections............ 65

3 Ordinary Diﬀerential Equations........................... 67

3.1 Global in Time Existence of Solutions of Ordinary

Diﬀerential Equations................................... 67

3.1.1 A necessary and suﬃcient condition for completeness

ofa vectorﬁeld ofone-sided type................... 67

3.1.2 A generalization to the inﬁnite-dimensional case . . . . . . 70

3.1.3 A necessary and suﬃcient condition for completeness

ofa vectorﬁeld oftwo-sided type................... 82

3.1.4 Some suﬃcient conditions . . . ...................... 84