The book presents an unusual and fresh way to organize a course for graduate or postgraduate students on global analysis of manifolds. The author starts with basic definitions and facts from the theory of smooth manifolds and continues with a discussion of differential operators on manifolds. He introduces the notion of a connection on a vector bundle and its curvature and gives more details on complex and symplectic manifolds and Riemannian spin geometry. The last two chapters contain a discussion of elliptic operators on compact manifolds (including necessary tools, e.g., the Fourier transform, the Sobolev theory, interior regularity and symbol calculus) and corresponding vanishing theorems. An unusual feature of the book is the fact that the explanation of many facts is not standard or they are discussed in a different order. The first and main tool introduced (even before a definition of a differential manifold) is the notion of a sheaf on a topological space. Differential operators are defined as linear operators on sheaves. Cohomology with values in a sheaf is then introduced and systematically used.

Reviewer:

vs