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Geometric properties of natural operators defined by the Riemann curvature tensor. (English) Zbl 1007.53001
Singapore: World Scientific. viii, 306 p. (2001).
The author considers natural operators associated to the pseudo-Riemannian curvature tensor: the Jacobi operator, the Szabó operator, the skew-symmetric curvature operator and other related operators, and discusses geometric conditions which are imposed when one of these operators has constant eigenvalues. The book is organized as follows. Chapter 1 entitled ‘Algebraic Curvature Tensors’ contains algebraic preliminaries. Chapter 2 deals with the skew-symmetric curvature operator. Chapter 3 deals with the Jacobi and Szabó operators. Chapter 4 discusses results from algebraic topology which are used in the previous chapters. An extensive bibliography is provided at the end of the book. As relationship between algebraic properties of the Riemannian curvature tensor and the underlying geometry is one of the central problems in differential geometry, the book under review is useful for researchers interested in pseudo-Riemannian geometry and its aplications.

MSC:
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53B21 Methods of local Riemannian geometry
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