A first course in geometric topology and differential geometry.

*(English)*Zbl 0873.53001
Boston: Birkhäuser. xii, 421 p. (1997).

The book under review is divided into eight chapters and for each chapter some exercises are given. The aim of this book is to give an introduction to geometric topology and differential geometry through the study of surfaces in the topological, polyhedral and smooth categories.

The topological study is presented in the chapters: topological surfaces (surfaces in \(\mathbb{R}^n\), properties of surfaces, connected sum and the classification of compact connected surfaces) and simplicial surfaces (simplicial surfaces, the Euler characteristic, simplicial curvature and the simplicial Gauss-Bonnet theorem).

The classical differential geometry of smooth surfaces is developed. The following subjects are treated: smooth surfaces, normal vectors, first fundamental form, length and area, isometries, Gaussian curvature, mean curvature, principal directions, theorema egregium and fundamental theorem of surfaces, geodesics. Smooth surfaces can be analyzed geometrically and topologically, and the famous Gauss-Bonnet theorem shows that these aspects are deeply related.

The author offers a sensitive discussion on these questions. The book also contains the differential geometry of curves in \(\mathbb{R}^3\). The author gives a synthesis of topological and differential methods in this text. Finally, hints for selected exercises are presented. Numerous illustrations and examples are presented, too.

In addition, the material in this book is an introduction to a number of branches of mathematics, and the author gives some references for each topic (algebraic topology, geometry topology, differential topology and modern differential geometry). The exposition is clear, nicely organized, and generally easy to read.

The topological study is presented in the chapters: topological surfaces (surfaces in \(\mathbb{R}^n\), properties of surfaces, connected sum and the classification of compact connected surfaces) and simplicial surfaces (simplicial surfaces, the Euler characteristic, simplicial curvature and the simplicial Gauss-Bonnet theorem).

The classical differential geometry of smooth surfaces is developed. The following subjects are treated: smooth surfaces, normal vectors, first fundamental form, length and area, isometries, Gaussian curvature, mean curvature, principal directions, theorema egregium and fundamental theorem of surfaces, geodesics. Smooth surfaces can be analyzed geometrically and topologically, and the famous Gauss-Bonnet theorem shows that these aspects are deeply related.

The author offers a sensitive discussion on these questions. The book also contains the differential geometry of curves in \(\mathbb{R}^3\). The author gives a synthesis of topological and differential methods in this text. Finally, hints for selected exercises are presented. Numerous illustrations and examples are presented, too.

In addition, the material in this book is an introduction to a number of branches of mathematics, and the author gives some references for each topic (algebraic topology, geometry topology, differential topology and modern differential geometry). The exposition is clear, nicely organized, and generally easy to read.

Reviewer: C.Apreutesei (Iaşi)

##### MSC:

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

57-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes |

53A05 | Surfaces in Euclidean and related spaces |