##
**Analysis, manifolds and physics. Part II: 92 applications.**
*(English)*
Zbl 0682.58002

Amsterdam etc.: North-Holland. xii, 449 p. (1989).

The book is a companion volume of the joint book of the authors and M. Dillard-Bleick [Analysis, manifolds and physics (1977; Zbl 0385.58001, 2nd ed. 1982; Zbl 0492.58001)]. However, some knowledge of differential geometry and distribution theory are sufficient to the reader of the present book.

With respect to the presentation of the material, the authors have had the great idea to develop in each section the concepts and theorems and then give applications to physics in forms of problems with solutions. In this way the reader is encouraged to work out his own answers and thus to pursue his studies beyond the confines of the present volume.

We now proceed with the presentation of the content of the book. The whole material is exposed in seven chapters: I. Review of fundamental notions of analysis, II. Differential calculus on Banach spaces, III. Differentiable manifolds, IV. Integration on manifolds, V. Riemannian manifolds. Kählerian manifolds, V. Bis. Connections on a principal fibre bundle, VI. Distributions. Throughout the book the interest of the authors in mathematical concepts with direct applications in supergravity is evident. That is why we find sections named as follows: graded algebras, Berezenian, Clifford algebras, supersmooth mappings, Berezin integration, graded bundles. On the other hand, Kaluza-Klein theories and gauge theories are rigorously presented in forms of problems. Finally, the reader will find informations on: homotopy groups, cohomology groups, Lie algebras, harmonic maps, Sobolev spaces, characteristic classes, etc. which have many applications not only to physics but to economics, biology and engineering.

The present book is very clear and well written. It should prove useful for graduate students and researchers in applications of differential geometry and distribution theory to physics and related fields.

With respect to the presentation of the material, the authors have had the great idea to develop in each section the concepts and theorems and then give applications to physics in forms of problems with solutions. In this way the reader is encouraged to work out his own answers and thus to pursue his studies beyond the confines of the present volume.

We now proceed with the presentation of the content of the book. The whole material is exposed in seven chapters: I. Review of fundamental notions of analysis, II. Differential calculus on Banach spaces, III. Differentiable manifolds, IV. Integration on manifolds, V. Riemannian manifolds. Kählerian manifolds, V. Bis. Connections on a principal fibre bundle, VI. Distributions. Throughout the book the interest of the authors in mathematical concepts with direct applications in supergravity is evident. That is why we find sections named as follows: graded algebras, Berezenian, Clifford algebras, supersmooth mappings, Berezin integration, graded bundles. On the other hand, Kaluza-Klein theories and gauge theories are rigorously presented in forms of problems. Finally, the reader will find informations on: homotopy groups, cohomology groups, Lie algebras, harmonic maps, Sobolev spaces, characteristic classes, etc. which have many applications not only to physics but to economics, biology and engineering.

The present book is very clear and well written. It should prove useful for graduate students and researchers in applications of differential geometry and distribution theory to physics and related fields.

Reviewer: A.Bejancu

### MSC:

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

58C50 | Analysis on supermanifolds or graded manifolds |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

58D30 | Applications of manifolds of mappings to the sciences |

81S40 | Path integrals in quantum mechanics |

83E50 | Supergravity |

91B99 | Mathematical economics |

92F05 | Other natural sciences (mathematical treatment) |