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Introduction to Differentiable Manifolds and Riemannian Geometry

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ISBN-10: 0121160513

ISBN-13: 9780121160517

Edition: 2nd 2003 (Revised)

Authors: William M. Boothby, William M. Boothby

List price: $119.00
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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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Book details

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 8.75" long x 1.00" tall
Weight: 1.298
Language: English

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
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
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