The geometry of curvature homogeneous pseudo-Riemannian manifolds. (English) Zbl 1128.53041

ICP Advanced Texts in Mathematics 2. London: Imperial College (ISBN 978-1-86094-785-8/hbk). xii, 376 p. (2007).
The main aim of the book under review is to study relationships between geometry of a pseudo-Riemannian manifold and algebraic properties of its curvature tensor.
It has six chapters and each chapter has an introductory section to recall preceding material.
The first chapter is devoted to a discussion of the geometric properties of the Riemannian curvature tensor and basic geometrical concepts to be used throughout the book. Moreover, algebraic curvature tensors are also studied that admit the same type of symmetries as the associated curvature tensor. This chapter also considers the Jacobi operator and fundamentals of spectral geometry.
In the second chapter, the author studies curvature homogeneous generalized plane wave manifolds, Nikčević manifolds, Dunn manifolds and \(k\)-curvature homogeneous manifolds.
Chapter 3 starts by studying examples which are not generalized plane wave manifolds as well as Lorentz manifolds, Walker manifolds of signature (2,2) and Fiedler manifolds. It also deals with questions of geodesic completeness and Ricci blowup.
The Nash embedding theorem is used to provide generators for the space of algebraic curvature tensors and algebraic covariant derivative tensors in Chapter 4. Jordan Osserman algebraic curvature tensors and Szabó covariant derivative algebraic tensors are discussed there. This chapter also considers questions in conformal geometry, Stanilov models, complex geometry.
In Chapter 5, the authors explore complex models that are both Osserman and complex Osserman and provides an almost complete classification of them except for some dimensions and ranks.
The last chapter begins with an introduction to Stanilov-Tsankov theory and studies Jacobi Tsankov manifolds, skew Tsankov manifolds, Stanilov-Videv manifolds and Jacobi Videv manifolds.
The author mentions that Chapters 5 and 6 are joint work with M. Brozos-Vázquez.


53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C30 Differential geometry of homogeneous manifolds
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry