The book "Analysis and Algebra of Dierentiable Manifolds" contains a

collection of solved problems on diverse topics enclosed in the theory of dier-

entiable manifolds and their geometry. Although the text does not cover the

theory to which the problems refer, it includes introductory explanations at the

beginning of each section. In addition, appropriate references containing the

theory needed to solve the problems are pointed out. At the end of each chap-

ter other related references and further readings are suggested. It also contains

66 gures.

The book is organized as follows. The rst three chapters comprehend all

the topics included in an standard course on dierentiable manifolds at an un-

dergraduate level, starting from the very denition of a dierentiable manifold,

passing through its tensor algebra, and ending with integration on manifolds

and de Rham cohomology. This rst part is of great value for both students

and teachers. The second part of the book is devoted to Lie groups, Rieman-

nian geometry and bre bundles, subjects that must be part of a dierential

geometer's background. The problems selected in this second part cover from

an advance undergraduate level to a Master level. The importance of these

chapters is that they treat topics for which signicant examples and problems

are very difficult to nd in the literature. This collection of exercises allows the

reader to explore and to handle many concepts which otherwise could remain

just as theoretical objects.

Reviewer:

Ignacio Luján

Affiliation:

Universidad Complutense de Madrid