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The geometry of Walker manifolds. (English) Zbl 1206.53039

A Walker structure on a differentiable manifold is given by a parallel distribution, totally isotropic with respect to a semi-Riemannian metric. Such structures were first studied by A. G. Walker in [Q. J. Math., Oxf. Ser. 20, 135–145 (1949; Zbl 0033.13101)] in connection with the problem of extending de Rham’s theorem to pseudo-Riemannian metrics. Since then, a lot of examples of Walker structures have appeared, which proved to be important in differential geometry and general relativity as well. The book under review gathers a great variety of such examples, coming from different \(G\)-structures (contact, (almost) complex, quaternion) compatible with a pseudo-Riemannian metric. But the interest of the authors is not directed at applications of Walker structures in relativity, but at the properties of the curvature tensor. Walker structures are viewed as a good environment where various kinds of algebraic curvature tensors, with various symmetries, commutating or spectral properties (of the Jacobi operator) do exist.
The contents of the book and the distribution of the material clearly shows this: 1. Basic algebraic notions (20 p.); 2. Basic geometrical notions (18 p.); 3. Walker structures (22 p. Half of this chapter is devoted to Riemannian extensions, including Osserman and Ivanov-Petrova manifolds); 4. Three-dimensional Lorentzian manifolds (20 p.) 5. Four-dimensional Walker manifolds (12 p., here the focus is on para-Hermitian geometry); 6. The spectral geometry of curvature tensor (13 p.); 7. Hermitian geometry (16 p.) 8. Special Walker manifolds (10 p.). Each chapter starts with a brief historical note, then lists a series of results, the majority of them without proof. The book ends with a comprehensive bibliography of 264 titles.
For the interested reader, this book is a very useful guide in the jungle of curvature-defined pseudo-Riemannian manifolds.

MSC:

53C20 Global Riemannian geometry, including pinching
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)

Citations:

Zbl 0033.13101
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