*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 (

[email protected]) 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.