Introduction to Differentiable Manifolds

Louis Auslander, Robert E. MacKenzie

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486471723
• eBook ISBN: 9780486158082
• Topic: Mathematics
• Subtopic: Differential Geometry
• Index: Mathematics

The first book to treat manifold theory at an introductory level, this text surveys basic concepts in the modern approach to differential geometry. The first six chapters define and illustrate differentiable manifolds, and the final four chapters investigate the roles of differential structures in a variety of situations.
Starting with an introduction to differentiable manifolds and their tangent spaces, the text examines Euclidean spaces, their submanifolds, and abstract manifolds. Succeeding chapters explore the tangent bundle and vector fields and discuss their association with ordinary differential equations. The authors offer a coherent treatment of the fundamental concepts of Lie group theory, and they present a proof of the basic theorem relating Lie subalgebras to Lie subgroups. Additional topics include fiber bundles and multilinear algebra. An excellent source of examples and exercises, this graduate-level text requires a solid understanding of the basic theory of finite-dimensional vector spaces and their linear transformations, point-set topology, and advanced calculus.

Table of contents

• Cover
• Title Page
• Dedication
• Copyright Page
• Preface
• Contents
• Chapter 1: Euclidean, Affine, and Differentiable Structure on Rn
• Chapter 2: Differentiable Manifolds
• Chapter 3: Projective Spaces and Projective Algebraic Varieties
• Chapter 4: The Tangent Bundle of a Differentiable Manifold
• Chapter 5: Submanifolds and Riemann Metrics
• Chapter 6: The Whitney Imbedding Theorem
• Chapter 7: Lie Groups and Their One-parameter Sub-groups
• Chapter 8: Integral Manifolds and Lie Subgroups
• Chapter 9: Fiber Bundles
• Chapter 10: Multilinear Algebra
• References
• Index