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This revised edition of William Boothby's classic introduction to differentiable manifolds includes updated references and indices and error corrections.

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List price: $119.00 Edition: 2nd Copyright year: 2003 Publisher: Elsevier Science & Technology Publication date: 9/8/2002 Binding: Paperback Pages: 400 Size: 6.00" wide x 9.00" long x 1.00" tall Weight: 1.298 Language: English

AuthorTable of Contents

Preface to the Revised Second Edition

Preface to the Second Edition

Preface to the First Edition

Introduction to Manifolds

Preliminary Comments on R[superscript n]

R[superscript n] and Euclidean Space

Topological Manifolds

Further Examples of Manifolds. Cutting and Pasting

Abstract Manifolds. Some Examples

Functions of Several Variables and Mappings

Differentiability for Functions of Several Variables

Differentiability of Mappings and Jacobians

The Space of Tangent Vectors at a Point of R[superscript n]

Another Definition of T[subscript a](R[superscript n])

Vector Fields on Open Subsets of R[superscript n]

The Inverse Function Theorem

The Rank of a Mapping

Differentiable Manifolds and Submanifolds

The Definition of a Differentiable Manifold

Further Examples

Differentiable Functions and Mappings

Rank of a Mapping, Immersions

Submanifolds

Lie Groups

The Action of a Lie Group on a Manifold. Transformation Groups

The Action of a Discrete Group on a Manifold

Covering Manifolds

Vector Fields on a Manifold

The Tangent Space at a Point of a Manifold

Vector Fields

One-Parameter and Local One-Parameter Groups Acting on a Manifold

The Existence Theorem for Ordinary Differential Equations

Some Examples of One-Parameter Groups Acting on a Manifold

One-Parameter Subgroups of Lie Groups

The Lie Algebra of Vector Fields on a Manifold

Frobenius's Theorem

Homogeneous Spaces

Tensors and Tensor Fields on Manifolds

Tangent Covectors

Covectors on Manifolds

Covector Fields and Mappings

Bilinear Forms. The Riemannian Metric

Riemannian Manifolds as Metric Spaces

Partitions of Unity

Some Applications of the Partition of Unity

Tensor Fields

Tensors on a Vector Space

Tensor Fields

Mappings and Covariant Tensors

The Symmetrizing and Alternating Transformations

Multiplication of Tensors

Multiplication of Tensors on a Vector Space

Multiplication of Tensor Fields

Exterior Multiplication of Alternating Tensors

The Exterior Algebra on Manifolds

Orientation of Manifolds and the Volume Element

Exterior Differentiation

An Application to Frobenius's Theorem

Integration on Manifolds

Integration in R[superscript n] Domains of Integration

Basic Properties of the Riemann Integral

A Generalization to Manifolds

Integration on Riemannian Manifolds

Integration on Lie Groups

Manifolds with Boundary

Stokes's Theorem for Manifolds

Homotopy of Mappings. The Fundamental Group

Homotopy of Paths and Loops. The Fundamental Group

Some Applications of Differential Forms. The de Rham Groups

The Homotopy Operator

Some Further Applications of de Rham Groups

The de Rham Groups of Lie Groups

Covering Spaces and Fundamental Group

Differentiation on Riemannian Manifolds

Differentiation of Vector Fields along Curves in R[superscript n]

The Geometry of Space Curves

Curvature of Plane Curves

Differentiation of Vector Fields on Submanifolds of R[superscript n]

Formulas for Covariant Derivatives

[down triangle, open subscript x subscript p] Y and Differentiation of Vector Fields

Differentiation on Riemannian Manifolds

Constant Vector Fields and Parallel Displacement

Addenda to the Theory of Differentiation on a Manifold

The Curvature Tensor

The Riemannian Connection and Exterior Differential Forms

Geodesic Curves on Riemannian Manifolds

The Tangent Bundle and Exponential Mapping. Normal Coordinates

Some Further Properties of Geodesics

Symmetric Riemannian Manifolds

Some Examples

Curvature

The Geometry of Surfaces in E[superscript 3]

The Principal Curvatures at a Point of a Surface

The Gaussian and Mean Curvatures of a Surface

The Theorema Egregium of Gauss

Basic Properties of the Riemann Curvature Tensor

Curvature Forms and the Equations of Structure

Differentiation of Covariant Tensor Fields

Manifolds of Constant Curvature

Spaces of Positive Curvature

Spaces of Zero Curvature

Spaces of Constant Negative Curvature

References

Index

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