Tensors and Riemannian Geometry: With Applications to Differential Equations

Description

Examples 2.3.1 and 2.3.2 (PDF pp. 45-49) are very illustrative regarding free-body and Kepler motion's symmetries and, via Noether's theorem, conservation laws.

Maxwell equations / Lorentz group on PDF pp. 123ff.

p. v (PDF p. 3): "F. Klein (42) has underscored that what is called in physics the special theory of relativity is, in fact, a theory of invariants of the Lorentz group."

Bp. des Lauriers's dissertation, Sur les systèmes différentiels du second ordre qui admettent un groupe continu fini de transformations (MathSciNet review), mentions Riemann and tensors, too.

This book is based on the experience of teaching the subject by the author in Russia, France, South Africa and Sweden. The author provides students and teachers with an easy to follow textbook spanning a variety of topics on tensors, Riemannian geometry and geometric approach to partial differential equations. Application of approximate transformation groups to the equations of general relativity in the de Sitter space simplifies the subject significantly.

Even though this book has Tensors and Riemannian geometry as its main title, it is not a book about geometry. The main theme of the book is implied in the subtitle, With applications to differential equations. If you are looking for a text about tensors and their applications to Riemannian geometry, the reviewer recommends that you try other sources [for example, J. G. Simmonds, A brief on tensor analysis, second edition, Undergrad. Texts Math., Springer, New York, 1994; MR1247707; P. Petersen V, Riemannian geometry, second edition, Grad. Texts in Math., 171, Springer, New York, 2006; MR2243772]. The book under review is designed for graduate courses in differential equations and mathematical modeling via Riemannian geometric methods, with tensors as major tools.
The book is divided into three parts. The first part reviews some basics of tensors and Riemannian spaces, as well as the conservative laws. The second part introduces the Riemannian spaces of second-order equations, and the last part is about the theory of relativity. It is less like a traditional text and more like seminar-lecture notes. The text focuses more on development of concepts and methods rather than on depth and details of the results. It is not easy to cover these topics in a short introduction course. The book does an excellent job of choosing related materials and presenting them with good examples in very concise ways that make the topics more accessible. The reviewer especially likes the historical comments and references that appear in the book, and wishes to have seen more. They are definitely helpful for the reader to learn about the historical perspective, and they make the presentation much more interesting. Reviewed by Jie Yang