Differential Geometry and Topology: With a View to Dynamical Systems

Keith Burns and Marian Gidea

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

MAA Review
Table of Contents

[Reviewed by

William J. Satzer

, on

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