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Publisher:

Chapman&Hall/CRC

Publication Date:

2005

Number of Pages:

389

Format:

Hardcover

Series:

Studies in Advanced Mathematics

Price:

89.95

ISBN:

1-58488-253-0

Category:

Textbook

[Reviewed by , on ]

William J. Satzer

10/23/2005

*Differential Geometry and Topology: With a View to Dynamical Systems* is an introduction to differential topology, Riemannian geometry and differentiable dynamics. The authors' intent is to demonstrate the strong interplay among geometry, topology and dynamics. The modern theory of dynamical systems depends heavily on differential geometry and topology as, illustrated, for example, in the extensive background section included in Abraham and Marsden's *Foundations of Mechanics*. On the other hand, dynamical systems have provided both motivation and a multitude of non-trivial applications of the powerful tools of differential geometry and topology. Indeed, the connections are deep, going back to the groundbreaking work of Henri Poincaré.

This book begins with the basic theory of differentiable manifolds and includes a discussion of Sard's theorem and transversality. The authors then consider vector fields on manifolds together with basic ideas of smooth and discrete dynamical systems. In a single section they discuss hyperbolic fixed points, the stable manifold theorem, and the Hartman-Grobman theorems for diffeomorphisms and for flows.

Succeeding chapters address Riemannian geometry (metrics, connections and geodesics), curvature, tensors and differential forms, singular homology and De Rham cohomology. An extensive chapter on fixed points and intersection numbers includes discussions of the Brouwer degree, Lefschetz number, Euler characteristic and versions of the Gauss-Bonnet theorem. The final two chapters address Morse theory and hyperbolic systems. Here, the authors present the important example of the gradient flow, as well as the Morse inequalities and homoclinic points via the Smale horseshoe.

The authors of this book treat a great many topics very concisely. The writing is clear but rather dry, marked by long sequences of theorem-proof-remark. One does not get much sense of context, of the strong connections between the various topics or of their rich history. One wishes for more concrete examples and exercises. Prerequisites include at least advanced calculus and some topology (at the level of Munkres' book). This book could be used as a text for a graduate course if the instructor filled in additional examples, exercises and discussion of context and connections.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

MANIFOLDS

Introduction

Review of topological concepts

Smooth manifolds

Smooth maps

Tangent vectors and the tangent bundle

Tangent vectors as derivations

The derivative of a smooth map

Orientation

Immersions, embeddings and submersions

Regular and critical points and values

Manifolds with boundary

Sard's theorem

Transversality

Stability

Exercises

VECTOR FIELDS AND DYNAMICAL SYSTEMS

Introduction

Vector fields

Smooth dynamical systems

Lie derivative, Lie bracket

Discrete dynamical systems

Hyperbolic fixed points and periodic orbits

Exercises

RIEMANNIAN METRICS

Introduction

Riemannian metrics

Standard geometries on surfaces

Exercises

RIEMANNIAN CONNECTIONS AND GEODESICS

Introduction

Affine connections

Riemannian connections

Geodesics

The exponential map

Minimizing properties of geodesics

The Riemannian distance

Exercises

CURVATURE

Introduction

The curvature tensor

The second fundamental form

Sectional and Ricci curvatures

Jacobi fields

Manifolds of constant curvature

Conjugate points

Horizontal and vertical sub-bundles

The geodesic flow

Exercises

TENSORS AND DIFFERENTIAL FORMS

Introduction

Vector bundles

The tubular neighborhood theorem

Tensor bundles

Differential forms

Integration of differential forms

Stokes' theorem

De Rham cohomology

Singular homology

The de Rham theorem

Exercises

FIXED POINTS AND INTERSECTION NUMBERS

Introduction

The Brouwer degree

The oriented intersection number

The fixed point index

The Lefschetz number

The Euler characteristic

The Gauss-Bonnet theorem

Exercises

MORSE THEORY

Introduction

Nondegenerate critical points

The gradient flow

The topology of level sets

Manifolds represented as CW complexes

Morse inequalities

Exercises

HYPERBOLIC SYSTEMS

Introduction

Hyperbolic sets

Hyperbolicity criteria

Geodesic flows

Exercises

References

Index

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