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A Short Course in Differential Geometry and Topology
A.T. Fomenko and A.S. Mishchenko

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A Short Course in Differential Geometry and Topology

A.T . Fomenko and A.S. Mishchenko Faculty of Mechanics and Mathematics, Moscow State University

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Cambridge Scientific Publishers


2009 Cambridg e Scientific Publisher s Cover design: Clare Turner All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without prior permission in writing from the publisher. Britis h Librar y Cataloguin g i n Publicatio n Dat a A catalogue record for this book has been requested Librar y o f Congres s Cataloguin g i n Publicatio n Dat a A catalogue record has been requested ISBN 978-1-904868-32-3 Cambridge Scientific Publishers Ltd P.O. Box 806 Cottenham, Cambridge CB24 8RT UK www.cambridgescientificpublishers.com Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne


Contents
Preface Introduction to Differential Geometry 1.1 Curvilinear Coordinate Systems. Simplest Examples 1.1.1 Motivation 1.1.2 Cartesian and curvilinear coordinates 1.1.3 Simplest examples of curvilinear coordinate systems . . 1.2 The Length of a Curve in Curvilinear Coordinates 1.2.1 The length of a curve in Euclidean coordinates 1.2.2 The length of a curve in curvilinear coordinates 1.2.3 Concept of a Riemannian metric in a domain of Euclidean space 1.2.4 Indefinite metrics 1.3 Geometry on th e Sphere and Plane 1.4 Pseudo-Sphere and Lobachevskii Geometry Genera l Topolog y 2.1 Definitions and Simplest Properties of Metric and Topological Spaces 2.1.1 Metric spaces 2.1.2 Topological spaces 2.1.3 Continuous mappings 2.1.4 Quotient topology 2.2 Connectedness. Separation Axioms 2.2.1 Connectedness 2.2.2 Separation axioms 2.3 Compact Spaces 2.3.1 Compact spaces 2.3.2 Properties of compact spaces 2.3.3 Compact metric spaces 2.3.4 Operations on compact spaces ix 1 1 1 3 7 10 10 12 15 18 20 25 39 39 39 40 42 44 45 45 47 49 49 50 51 51


vi 2.4 Functional Separability. Partition of Unity 2.4.1 Functional separability 2.4.2 Partition of unity 3

CONTENTS 51 52 54

Smoot h Manifolds (General Theory ) 57 3.1 Concept of a Manifold 59 3.1.1 Main definitions 59 3.1.2 Functions of change of coordinates. Definition of a smooth manifold 62 3.1.3 Smooth mappings. Diffeomorphism 65 3.2 Assignment of Manifolds by Equations 68 3.3 Tangent Vectors. Tangent Space 72 3.3.1 Simplest examples 72 3.3.2 General definition of tangent vector 74 3.3.3 Tangent space TPo(M) 75 3.3.4 Directional derivative of a function 76 3.3.5 Tangent bundle 79 3.4 Submanifolds 81 3.4.1 Differential of a smooth mapping 81 3.4.2 Local properties of mappings and the differential . . . . 84 3.4.3 Embedding of manifolds in Euclidean space 85 3.4.4 Riemannian metric on a manifold 87 3.4.5 Sard theorem 89 93 93 93 98 102 102 104 108 112 121 121 131 132 132

4 Smoot h Manifolds (Examples) 4.1 Theory of Curves on th e Plane and in th e ThreeDimensional Space 4.1.1 Theory of curves on the plane. Frenet formulae 4.1.2 Theory of spatial curves. Frenet formulae 4.2 Surfaces. First and Second Quadratic Forms 4.2.1 First quadratic form 4.2.2 Second quadratic form 4.2.3 Elementary theory of smooth curves on a hypersurface 4.2.4 Gaussian and mean curvatures of two-dimensional surfaces 4.3 Transformation Groups 4.3.1 Simplest examples of transformation groups 4.3.2 Matrix transformation groups 4.3.3 General linear group 4.3.4 Special linear group


CONTENTS 4.3.5 Orthogonal group 4.3.6 Unitary group and special unitary group 4.3.7 Symplectic compact and noncompact groups 4.4 Dynamical Systems 4.5 Classification of Two-Dimensional Surfaces 4.5.1 Manifolds with boundary 4.5.2 Orientable manifolds 4.5.3 Classification of two-dimensional manifolds 4.6 Two-Dimensional Manifolds as Riemann Surfaces of Algebraic Functions 5 Tensor Analysi s 5.1 General Concept of Tensor Field on a Manifold 5.2 Simplest Examples of Tensor Fields 5.2.1 Examples 5.2.2 Algebraic operations on tensors 5.2.3 Skew-symmetric tensors 5.3 Connection and Covariant Differentiation 5.3.1 Definition and properties of affine connection 5.3.2 Riemannian connections 5.4 Parallel Translation. Geodesies 5.4.1 Preparatory remarks 5.4.2 Equation of parallel translation 5.4.3 Geodesies 5.5 Curvature Tensor 5.5.1 Preparatory remarks 5.5.2 Coordinate definition of the curvature tensor 5.5.3 Invariant definition of the curvature tensor 5.5.4 Algebraic properties of the Riemannian curvature tensor 5.5.5 Some applications of the Riemannian curvature tensor 6 Homology Theor y 6.1 Calculus of Differential Forms. Cohomologies 6.1.1 Differential properties of exterior forms 6.1.2 Cohomologies of a smooth manifold (de Rham cohomologies) 6.1.3 Homotopic properties of cohomology groups 6.2 Integration of Exterior Forms 6.2.1 Integral of a differential form over a manifold

vii 133 134 137 140 149 150 151 153 163 173 173 177 177 180 183 189 189 195 198 198 199 201 210 210 210 211 212 215 219 220 220 225 227 231 231


6.2.2 6.3 Degree 6.3.1 6.3.2 6.3.3 6.3.4

Stokes formula of a Mapping and Its Applications Degree of a mapping Main theorem of algebra Integration of forms Gaussian mapping of a hypersurface

232 236 236 238 239 239

Simplest Variational Problem s of Riemannia n Geometr y 241 7.1 Concept of Functional. Extremal Functions. Euler Equation . . 241 7.2 Extremality of Geodesies 247 7.3 Minimal Surfaces 250 7.4 Calculus of Variations and Symplectic Geometry 253 Bibliography Index 267 269


Preface
A Short Course on Differential Geometr y and Topology by Professor A.T. Fomenko and Professor A.S. Mishchenko is based on the course taught at the Faculty of Mechanics and Mathematics of Moscow State University. It is intended for students of mathematics, mechanics and physics and also provides a useful reference text for postgraduates and researchers specialising in modern geometry and its applications. The course is structured in seven chapters and covers the basic material on general topology, nonlinear coordinate systems, theory of smooth manifolds, theory of curves and surfaces, transformation groups, tensor analysis and Riemannian geometry, theory of integration and homologies, fundamental groups and variational problems of Riemannian geometry. All of the chapters are highly illustrated and the text is supplemented by a large number of definitions, examples, problems, exercises to test th e concepts introduced. The material has been carefully selected and is presented in a concise manner which is easily accessible to students. Chapter 1: Introduction to Differential Geometry This chapter consists of four sections and includes definitions, examples, problems and illustrations to aid the reader. 1.1 Curvilinear Coordinate Systems. Simplest Examples 1.2 The Length of a Curve in Curvilinear Coordinates 1.3 Geometry on the Sphere and Plane 1.4 Pseudo-Sphere and Lobachevskii Geometry
Chapter 2: General Topology

This chapter is an introduction to topology and the development of topology. Topology is a field of mathematics which studies the properties of geometric that are not changed under a "deformation" or other transformations similar to deformations. General topology arose as a result of studying the most general properties of geometric spaces and their transformations related to the convergence and continuity properties.
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x

PREFACE

The chapter consists of four sections and includes examples, definitions, problems and illustrations. 2.1 Definitions and Simplest Properties of Metric and Topological Spaces 2.2 Connectedness. Separation Axioms 2.3 Compact Spaces 2.4 Functional Separability. Partition of Unity Chapter 3: Smooth Manifolds: General Theory This chapter covers the general theory of smooth manifolds, introduces the manifold as a special concept in geometry and includes definitions, examples and problems to test understanding. 3.1 Concept of a Manifold 3.2 Assignment of Manifolds in Equations 3.3 Tangent Vectors. Tangent Space 3.4 Submanifolds Chapter 4: Smooth Manifolds: Examples This chapter continues the study of smooth manifolds and focuses on examples. 4.1 Theory of Curves on the Plane and in the Three-Dimensional Space 4.2 Surfaces 4.3 Transformation Groups 4.4 Dynamical Systems 4.5 Classification of Two-Dimensional Surfaces 4.6 Two-Dimensional Manifolds on Riemann Surfaces Chapter 5: Tensor Analysis and Riemannian Geometry This chapter studies local properties of smooth manifolds and includes definitions, illustrations, exercises and examples throughout. 5.1 General Concept of Tensor Field on a Manifold 5.2 Simplest Examples of Tensor Fields 5.3 Connection and Covariant Differentiation 5.4 Parallel Translation. Geodesies 5.5 Curvature Tensor Chapter 6: Homology Theory This chapter focuses on the properties of manifolds on which other functions and mapping depend. Examples and illustrations are included throughout the chapter. 6.1 Calculus of Differential Forms 6.2 Integration of Exterior Forms 6.3 Degree of a Mapping and its Applications


PREFACE

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Chapter 7: Simplest Variational Problems of Riemannian Geometry This chapter begins with the general concept of functional and its variation and subsequent sections cover extremality of geodesies, minimal surfaces, calculus of variations and symplectic geometry. 7.1 Concept of Functional. Extremal Functions. Euler Equation 7.2 Extremality of Geodesies 7.3 Minimal Surfaces 7.4 Calculus of Variations and Symplectic Geometry A Short Course in Differential Geometr y and Topology is intended for students of mathematics, mechanics and physics and also provides a useful reference text for postgraduates and researchers specialising in modern geometry and its applications. Selected Problem s in Differential Geometr y and Topology A.T. Fomenko, A.S. Mischenko and Y.P. Solovyev ISBN: 978-1-904868-33-0 Cambridge Scientific Publishers 2008 is designed as an associated companion volume to A Short Course in Differential Geometry and Topology and is based on seminars conducted at the Faculty of Mechanics and Mathematics of Moscow State University for second and third year mathematics and mechanics students. The two volumes complement each other and can be used as a two volume set.


A Short Course in Differential Geometry and Topology
A.T. Fomenko and A.S. Mishchenko Moscow State University
Thi s volume is intended for graduates and research students in mathematics and physics. It covers general topology, nonlinear co-ordinate systems, theory of smooth manifolds, theory of curves and surfaces, transformation groups, tensor analysis and Riemannian geometry, theory of integration and homologies, fundamental groups and variational principles in Riemannian geometry. Th e text is presented in a form tha t is easily accessible to students and is supplemented by a large numbe r of examples, problems, drawings and appendices.

ABOU T TH E AUTHOR S Anatoly T. Fomenko is a full member of the Russian Academy of Sciences an d Professor, Head of Departmen t of Differential Geometry and Applications, Faculty of Mathematics and Mechanics at Moscow State University. He is a wellknown specialist and the autho r of fundamental results in th e fields of geometry, topology, multidimensional calculus of variations, Hamiltonia n mechanics an d compute r geometry. Alexander S.Mishchenko is Professor in the Departmen t of Higher Geometry and Topology at the Faculty of Mathematics and Mechanics, Moscow State University. He is a well-known author and specialist in the field of algebraic topology, symplectic topology, functional analysis, differential equations and applications.

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CAMBRIDG E SCIENTIFI C PUBLISHERS