Differential Geometry and Topology - With a View to Dynamical Systems
This book offers a nice introduction to major topics in differential geometry and differential topology and their applications in the theory of dynamical systems. It starts with a chapter on manifolds (including the Sard theorem), followed by a discussion of vector fields, the Lie derivative and Lie brackets, and discrete and smooth dynamical systems. The following chapters treat Riemannian manifolds, affine and Levi-Civita connections, geodesics, curvatures, Jacobi fields and conjugate points and the geodesic flow. The chapter on tensors and differential forms includes integration of differential forms, Stokes theorem and a discussion of the de Rham and singular homology. Chapter 7 contains a description of the Brouwer degree, intersection numbers, Euler characteristics, and the Gauss-Bonnet theorem. Chapter 8 treats Morse theory and the final chapter discusses hyperbolic dynamical systems and geodesic flows. The book is nicely written and understandable, with many illustrations and intuitive comments. It is very suitable as an introduction to the field.