Differentiable Manifolds - ISBN:0030754852

Differentiable Manifolds

Date: 21 April 2011, 05:12

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From Preface:
The object of this book is to provide a text for a one-semester course in
the subject, essentially following the first fourteen paragraphs of the
CUPM recommendations for algebraic topology. The present text can be
used either before or after the author's Homology Theory, published by
Holden-Day, Inc., in 1966, to provide a one-year course in algebraic topology.
If the academic year is divided into three quarters, the author's Cohomology
Theory, published by Markham Publishing Company in 1968, can be used
to supplement these two books in the third quarter.
For pedagogical reasons, the text is restricted to finite-dimensional
manifolds, although the fundamental materials can be established at no
extra cost for manifolds modeled on Banach or Hilbert spaces rather than
finite-dimensional Euclidean spaces. Extensions to infinite-dimensional
manifolds are left to the reader as exercises. Intelligibility is always preferred
to brevity, and unnecessary generalizations are avoided as much as possible.
Table of Contents
Preface
Chapter I Differentiable Manifolds
1. Topological Manifolds
2. Differentiable Structures
3. Differentiable Functions
4. Tangent and Cotangent Spaces
5. Differential of Smooth Map
6. Vector Bundles
Chapter II Differential Forms
1. Grassmann Algebras
2. Differential Forms
3. Exterior Differentiation
4. De Rham Cohomology Groups
5. Induced Transformations
6. Poincare's Lemma
Chapter III Riemannian Manifolds
1. Inner Products
2. Riemannian Structures
3. Riemannian Metric
4. Riemannian Connection
5. Geodesics
6. Convex Neighborhoods
Chapter IV De Rham's Theorem
1. Singular Homology Groups
2. Real Singular Cohomology Groups
3. De Rham's Theorem
Bibliography
Index

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