Customer Reviews

Differential Geometry and Topology: With a View to Dynamical Systems (Studies in Advanced Mathematics)

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A very clear and very entertaining book for a course on differential geometry and topology (with a view to dynamical systems).

First let me remark that talking about content, the book is very good. Each of the 9 chapters of the book offers intuitive insight while developing the main text and it does so without lacking in rigor. The first 6 chapters (which deal with manifolds, vector fields and dynamical systems, Riemannian metrics, Riemannian connections and geodesics, curvature and tensors and differential forms) make up an introduction to dynamical systems and Morse theory (the subject of chapter 8). Chapter 7 is devoted to fixed points and intersection numbers. The last chapter is an introduction to hyperbolic systems.

This enjoyable and highly instructive book contains a large number of examples and exercises. It is an incredible help to those trying to learn dynamical systems (and not only). It teaches all the differential geometry and topology notions that somebody needs in the study of dynamical systems.
The authors, without making use of a pedantic formalism, emphasize the connection of important ideas via examples. It completely enhanced my knowledge on the subject and took me to a higher level of understanding.

It was a great pleasure to read the book Differential Geometry and Topology With a View to Dynamical Systems by Keith Burns and Marian Gidea. The topic of manifolds and its development, typically considered as very abstract and difficult, becomes for the reader of this outstanding book tangible and familiar. This joyful aspect of the book was achieved by the authors by setting the advanced material of differential geometry and topology as if on a mobile bridge or a crossroad that associates a(n) (primarily) unfamiliar abstract part of the text with elementary math theories. The latter pedagogical approach was mostly carried out through carefully prepared examples, in which, for essentially abstract structures and mathematical topics, well known familiar elementary settings serve as obvious motivations, which make the transition to a higher level of an abstraction smooth. Nevertheless, the scope of the main topic in this book, differential geometry and topology, is pretty far advanced. Besides the basic theory, centered around analytical properties of manifolds (mostly endowed with additional, in particular Riemannian, structures and vector or tensor fields defined on them) and their applications, it also provides a good introductory approach to some deeper topics of differential topology such as Fixed Points theory, Morse theory, and hyperbolic systems throughout the rest of the book.

The main stream of the applications that always follow or motivate the theoretical context is dynamical systems. Excellent examples reveal the close ties of this beautiful mathematical theory with common problems in theoretical physics, classical and fluid mechanics, field theory, and, most importantly, the theory of general relativity.

The book by Burns and Gidea is also be strongly recommended for those readers who wish to enhance their mathematical tools to make possible a deeper insight into these fascinating physical theories.

Jerzy K. Filus