Smooth manifolds and fibre bundles with applications to theoretical physics (Q2832135)

From the Preface: ``The intention of this book is to provide the necessary mathematical background from the theory of smooth manifolds, fibre bundles, Lie groups and differential geometry for students interested in classical mechanics and the general theory of relativity, and for students interested in the wide range of applications in diverse areas of mathematics. Although these mathematical subjects become more and more important in the recent developments of theoretical physics, it is hard to find books in physics with a self-contained presentation of the mathematical foundation needed in the physical theory. The purpose of this book is to fill this gap.''NEWLINENEWLINEThe book is the result of many years of dedicated work by the author lecturing on the material and writing it up as a manuscript for a book. In May 2013, he had a stroke preventing him from taking care of the final preparation of the manuscript, but luckily enough some of his colleagues assisted in the process of having this valuable book published.NEWLINENEWLINEThe book contains twelve chapters, an appendix with various preliminaries from homotopy theory and the theory of topological vector spaces, and a bibliography.NEWLINENEWLINEChapter 1 recalls briefly notions from differential geometry of curves and surfaces in 3-dimensional Euclidean space. Chapter 2 is an adequate introduction to smooth manifolds and vector bundles. Chapter 3 follows up with basic theory of vector fields and differential equations on smooth manifolds. The next three chapters give comprehensive accounts of the theory of tensors (Chapter 4), the theory of differential forms (Chapter 5) and integration on manifolds (Chapter 6). Chapter 7 is a very fine account of metric and symplectic structures. The next two chapters are delightful introductions to Lie groups and their Lie algebras (Chapter 8) and to group actions on smooth manifolds (Chapter 9), with applications among others to Lagrangian systems and gravitational central fields. The comprehensive Chapter 10 covers in adequate detail the important theory of fibre bundles. Chapter 11 discusses isometric immersions (between pseudo-Riemannian manifolds) and the second fundamental form. The material is applied in the description of various spacetimes and cosmological models. The final Chapter 12 is a good introduction to the theory of jet bundles, with application to the calculus of variations.NEWLINENEWLINEThe book is very carefully written paying good attention to necessary details. This surely makes the book valuable to students of physics who insist on stringency in the mathematical details of physical theories.