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**Differential geometry: Cartan’s generalization of Klein’s Erlangen program. Foreword by S. S. Chern.**
*(English)*
Zbl 0876.53001

Graduate Texts in Mathematics. 166. Berlin: Springer. xix, 421 p. (1997).

This is the first book in the literature that fully explains why differential geometry is the study of a connection on a principal bundle.

In the first chapter, the author shows how the differential calculus lives on a smooth manifold. In particular, there are introduced and discussed submanifolds, fibre bundles, vector fields, differential forms, etc. The next chapter is devoted to foliations. The author starts with the one-dimensional foliation, which consists of the integral curves of a nonzero vector field, and continues with the study of distributions and their integrability conditions. Here it is very clearly explained that the modern study of integrable distributions is just the theory of foliations.

The main purpose of Chapter 3 is to discuss a non-Abelian analog of the elementary fundamental theorem of calculus. The Maurer-Cartan form and the Darboux derivative are the basic tools to achieve both the local and global version of the fundamental theorem of calculus on a manifold. Further there is given a characterization of a Lie group in terms of the Maurer-Cartan form, and it is fully presented the correspondence between Lie groups and Lie algebras.

A Klein geometry is a pair \((G,H)\), where \(G\) is a Lie group and \(H\subset G\) is a closed subgroup such that \(G/H\) is connected. Examples of planar Klein geometries are given in the first section of Chapter 4. Then the chapter is dealing with the general theory of Klein geometries including the tangent bundle of a Klein geometry and the gauge view of Klein geometries. Chapter 5 is concerned with the study of Cartan geometry as a generalization of Klein geometry. There are presented both the local and global definition of a Cartan geometry, followed by a study of some concepts related to this geometry including tangent bundle, curvature function, tensors, etc. By means of the geometric orientation in Cartan geometries there are classified locally Klein geometries. The last two sections are dealing with the classification of Cartan space forms and its applications to symmetric spaces. The last three chapters are devoted to the study of some special Cartan geometries, namely: Riemannian geometry, Möbius geometry, and projective geometry. Five appendices that cover several concepts discussed throughout the book are added at the end of the book.

Finally, we should say that the book is well written and technically correct. Every chapter begins with an informal discussion, in which the author explains to the reader what is really going on and why the topic is important and interesting. The rigorous treatment is supplemented with helpful remarks and a wide range of exercises. The book is recommended to graduate students and researchers interested in differential geometry.

In the first chapter, the author shows how the differential calculus lives on a smooth manifold. In particular, there are introduced and discussed submanifolds, fibre bundles, vector fields, differential forms, etc. The next chapter is devoted to foliations. The author starts with the one-dimensional foliation, which consists of the integral curves of a nonzero vector field, and continues with the study of distributions and their integrability conditions. Here it is very clearly explained that the modern study of integrable distributions is just the theory of foliations.

The main purpose of Chapter 3 is to discuss a non-Abelian analog of the elementary fundamental theorem of calculus. The Maurer-Cartan form and the Darboux derivative are the basic tools to achieve both the local and global version of the fundamental theorem of calculus on a manifold. Further there is given a characterization of a Lie group in terms of the Maurer-Cartan form, and it is fully presented the correspondence between Lie groups and Lie algebras.

A Klein geometry is a pair \((G,H)\), where \(G\) is a Lie group and \(H\subset G\) is a closed subgroup such that \(G/H\) is connected. Examples of planar Klein geometries are given in the first section of Chapter 4. Then the chapter is dealing with the general theory of Klein geometries including the tangent bundle of a Klein geometry and the gauge view of Klein geometries. Chapter 5 is concerned with the study of Cartan geometry as a generalization of Klein geometry. There are presented both the local and global definition of a Cartan geometry, followed by a study of some concepts related to this geometry including tangent bundle, curvature function, tensors, etc. By means of the geometric orientation in Cartan geometries there are classified locally Klein geometries. The last two sections are dealing with the classification of Cartan space forms and its applications to symmetric spaces. The last three chapters are devoted to the study of some special Cartan geometries, namely: Riemannian geometry, Möbius geometry, and projective geometry. Five appendices that cover several concepts discussed throughout the book are added at the end of the book.

Finally, we should say that the book is well written and technically correct. Every chapter begins with an informal discussion, in which the author explains to the reader what is really going on and why the topic is important and interesting. The rigorous treatment is supplemented with helpful remarks and a wide range of exercises. The book is recommended to graduate students and researchers interested in differential geometry.

Reviewer: A.Bejancu (Iaşi)

### MSC:

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

53C05 | Connections (general theory) |

55R10 | Fiber bundles in algebraic topology |