Lectures on the geometry of manifolds / Liviu I. Nicolaescu

This book is a recommendable exposition of differential geometry. The text begins with preparatory material, such as the notion of smooth manifolds, vector bundles, and aspects of tensor calculus. Chapter 1 introduces the concept of smooth manifolds and the notion of Lie groups. Chapter 2 deals with basic constructions on manifolds. Chapter 3 contains techniques of the calculus on manifolds, the Lie derivative, connections on vectors bundles, integration on manifolds as well as a whole section on representation theory of compact Lie groups. Chapter 4 deals with Riemannian geometry including the geometry of submanifolds and concludes with the Gauss-Bonnet theorem. A brief chapter containing basic techniques of the calculus of variations (Chapter 5) is followed by a discussion of the fundamental group and covering spaces (Chapter 6). Chapter 7 presents topological material describing DeRham cohomology, Poincaré duality, intersection theory, symmetric spaces, and Čech cohomology, whereas Chapter 8 introduces the tool of characteristic classes. The topic of Chapter 9 is the study of elliptic equations on manifolds starting with basic notions on partial differential operators and concluding with Fredholm, spectral, and Hodge theory. The book concludes with an introductory chapter on Dirac operators including fundamental examples and the spin c case. The book is marked by its clear presentation, contains many exercises and is illustrated by numerous detailed examples. (Zentralblatt)

Bibliographie p. 469-473. Index