Symmetries and Curvature in General Relativity

This is a book describing the mathematics, and primarily the geometry, needed for the general theory of relativity. The first five chapters describe very basic notions. The exposition starts with groups, some linear algebra (including the Jordan canonical form of an endomorphism) and an introduction to the notion of Lie algebra, followed by metric and topological spaces, their various properties and finally the notion of a fundamental group and a covering space. The next part is devoted to differentiable manifolds and basic structures on them. Finally, the last section of this introductory part is devoted to Lie groups. This part, in fact, can be read by anybody who intends to learn the above mentioned topics. It should be mentioned that this introductory text is very nicely written.

The more specialized second part of the book starts with a chapter on the Lorentz group. It provides a lot of information about this group in compact form. It would hardly be possible to find all these facts together in one source elsewhere. Then the notion of a space-time (a 4-dimensional manifold endowed with a pseudo-Riemannian metric with signature + + + –) appears and the rest of the book deals with these space-time manifolds. Here, there are many results concerning this special branch of differential geometry: firstly the algebraic classification of space-time manifolds (above all the Petrov classification), than an investigation of the holonomy groups of these manifolds, an investigation of connections and the corresponding curvature tensors on these manifolds and finally symmetries. Isometries, homotheties, conformal symmetries, projective symmetries and curvature collineations are all studied. This second part, which represents the core of the book, will be attractive for physicists, who can revise their knowledge in a more mathematical form, but also for mathematicians working in differential geometry whose field is a little aside from space-time manifolds. Because of the first introductory part, the book is to a great extent self-contained and consequently can be recommended for students, possibly even undergraduate students with enough endurance. In the bibliography there are many references for further reading and for the study of relativity theory.

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