An Introductory Course on Differentiable Manifolds

by Siavash Shahshahani

Overview

Based on author Siavash Shahshahani's extensive teaching experience, this volume presents a thorough, rigorous course on the theory of differentiable manifolds. Geared toward advanced undergraduates and graduate students in mathematics, the treatment's prerequisites include a strong background in undergraduate mathematics, including multivariable calculus, linear algebra, elementary abstract algebra, and point set topology. More than 200 exercises offer students ample opportunity to gauge their skills and gain additional insights.
The four-part treatment begins with a single chapter devoted to the tensor algebra of linear spaces and their mappings. Part II brings in neighboring points to explore integrating vector fields, Lie bracket, exterior derivative, and Lie derivative. Part III, involving manifolds and vector bundles, develops the main body of the course. The final chapter provides a glimpse into geometric structures by introducing connections on the tangent bundle as a tool to implant the second derivative and the derivative of vector fields on the base manifold. Relevant historical and philosophical asides enhance the mathematical text, and helpful Appendixes offer supplementary material.

Product Details

ISBN-13:	9780486807065
Publisher:	Dover Publications
Publication date:	08/17/2016
Series:	Aurora: Dover Modern Math Originals
Edition description:	First Edition, First
Pages:	368
Product dimensions:	5.90(w) x 9.00(h) x 0.60(d)

About the Author

Siavash Shahshahani studied at Berkeley with Steve Smale and received his PhD in 1969, after which he held positions at Northwestern and the University of Wisconsin, Madison. From 1974 until his 2012 retirement he was mainly at Sharif University of Technology in Tehran, Iran, where he helped develop a strong mathematics program.

Table of Contents

Preface 1

Part I Pointwise 5

Chapter 1 Multilinear Algebra 7

A Dual Space 7

B Tensors 8

C Anti-symmetric Tensors 13

D Real Linear Spaces 17

E Product Structure 19

Exercises 24

Part II Local 27

Chapter 2 Vector Fields: Local Theory 29

A Tangent Space 29

B Vector Fields and Differential Equations 33

C Vector Fields as Operators 44

D Lie Bracket of Vector Fields 51

Exercises 60

Chapter 3 Tensor Fields: Local Theory 65

A Basic Constructions 65

B Pointwise Operations 68

C Exterior Derivative 72

D Lie Derivative 77

E Riemannian Metrics 81

Exercises 86

Part III Global 91

Chapter 4 Manifolds, Tangent Bundle 93

A Topological Manifolds 93

B Smooth Manifolds 102

C Smooth Structures 104

D The Tangent Bundle 109

Appendix 120

Exercises 122

Chapter 5 Mappings, Submanifolds and Quotients 125

A Submanifolds 125

B Immersions, Submersions and Embeddings 132

C Quotient Manifolds 137

D Covering Spaces 142

Exercises 150

Chapter 6 Vector Bundles and Fields 155

A Basic Constructions 155

B Vector Fields: Globalization 164

C Differential Forms: Globalization 169

D Riemannian Metrics 175

E Plane Fields 181

Exercises 195

Chapter 7 Integration and Cohomology 203

A Manifolds with Boundary 203

B Integration on Manifolds with Boundary 209

C Stokes Theorem 217

D De Rham Cohomology 222

E Top-dimensional Cohomology and Applications 232

Exercises 242

Chapter 8 Lie Groups and Homogeneous Spaces 251

A Continuous Groups 251

B The Lie Algebra of a Lie Group 261

C Homogeneous Spaces 271

Exercises 279

Part IV Geometric Structures 285

Chapter 9 An Introduction to Connections 287

A The Geography of the Double Tangent Bundle 287

B Descent of the Second Derivative 295

C Covariant Derivative 303

D Cutvature and Torsion 310

E Newtonian Mechanics 321

Exercises 330

Appendix I The Exponential of a Matrix 337

Appendix II Differential Calculus in Normed Space 341

Bibliography 349

List of symbols 351

Index 353