An Introduction to Differential Manifolds

by Barden, Dennis; Thomas, Charles B.

• ISBN13: 9781860943553
• ISBN10: 1860943551
• Edition: Reprint
• Format: Paperback
• Copyright: 2003-03-01
• Publisher: Imperial College Pr

Summary

This invaluable book, based on the many years of teaching experience of both authors, introduces the reader to the basic ideas in differential topology. Among the topics covered are smooth manifolds and maps, the structure of the tangent bundle and its associates, the calculation of real cohomology groups using differential forms (de Rham theory), and applications such as the PoincariHopf theorem relating the Euler number of a manifold and the index of a vector field. Each chapter contains exercises of varying difficulty for which solutions are provided. Special features include examples drawn from geometric manifolds in dimension 3 and Brieskorn varieties in dimensions 5 and 7, as well as detailed calculations for the cohomology groups of spheres and tori.

Table of Contents

Preface	v
Differential Manifolds and Differentiable Maps	1	(22)
Differential Manifolds: Definitions and Examples	1	(8)
Differentiable Maps: Definitions and Examples	9	(2)
Submanifolds and the Inverse Function Theorem	11	(4)
Bump Functions and the Embedding Theorem	15	(3)
Partitions of Unity	18	(3)
Exercises for Chapter 1	21	(2)
The Derivatives of Differentiable Maps	23	(20)
Tangent Vectors	23	(2)
The Tangent Space	25	(1)
The Tangent and Cotangent Bundles	26	(5)
Whitney's Embedding Theorem Revisited	31	(2)
Vector Fields and 1-forms on Euclidean Spaces	33	(4)
The Lie Bracket of Vector Fields	37	(1)
Integral Curves and 1-parameter Groups of Diffeomorphisms	38	(3)
Exercises for Chapter 2	41	(2)
Fibre Bundles	43	(22)
Coordinate Bundles	43	(7)
Vector Bundles	50	(7)
Riemannian Metrics
Applications	57	(7)
Exercises for Chapter 3	64	(1)
Differential Forms and Integration	65	(12)
Forms on Vector Spaces	66	(3)
Forms on Manifolds	69	(3)
The Orientation of Manifolds	72	(2)
Integration of m-forms on Oriented m-manifolds	74	(2)
Exercises for Chapter 4	76	(1)
The Exterior Derivative	77	(18)
The Exterior Derivative on Rm	78	(2)
The Exterior Derivative on Manifolds	80	(1)
Manifolds with Boundary	81	(4)
Stokes' Theorem	85	(3)
Bubbling Forms	88	(4)
Exercises for Chapter 5	92	(3)
De Rham Cohomology	95	(24)
Basic Definitions	96	(2)
Cochain Maps and Cochain Homotopies	98	(3)
The Poincare Lemma	101	(4)
The Mayer-Vietoris Sequence	105	(5)
The de Rham Groups of Spheres	110	(1)
The de Rham Groups of Tori	111	(4)
Homology and Submanifolds	115	(1)
Exercises for Chapter 6	116	(3)
Degrees, Indices and Related Topics	119	(34)
The Degree of a Mapping	120	(3)
Linking Numbers	123	(2)
The Index of a Vector Field	125	(5)
The Gauss Map	130	(5)
Morse Functions	135	(6)
The Euler Number	141	(4)
Handle Decompositions	145	(6)
Exercises for Chapter 7	151	(2)
Lie Groups	153	(32)
Lie Groups	153	(3)
Lie Algebras	156	(2)
The Exponential Map	158	(4)
Maximal Tori and Cohomology	162	(2)
Exercises for Chapter 8	164	(1)
A Rapid Course in Differential Analysis
Prerequisites for a Course on Differential Manifolds	165	(1)
Metric Spaces	165	(3)
Contraction Mappings	168	(3)
Differential Analysis	171	(5)
The Inverse Function Theorem	176	(3)
Sard's Theorem	179	(3)
Modules and Algebras	182	(1)
Tensor Products	183	(1)
Exterior Products	184	(1)
Solutions to the Exercises	185	(20)
Guide to the Literature	205	(4)
Literature References	209	(2)
General References	211	(4)
Index	215