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On some class of pseudosymmetric warped products. (English) Zbl 1053.53017
The class of warped product manifolds, for short warped products, is an extension of the class of products of semi-Riemannian manifolds. Many well-known space-times of general relativity, i.e. solutions of the Einstein equations, are warped products. The warped product \(\overline M\times_F\widetilde N\) of \((\overline M,\overline g)\), \(\dim\overline M \geq 1\), and \((\widetilde N,\widetilde g)\), \(\dim\widetilde N\geq 1\), \(n = \dim\overline M + \dim\widetilde N\geq 4\), is said to be a space-time of Robertson-Walker type if it has signature \((1, n - 1)\) and at least one of the manifolds \((\overline M,\overline g)\) and \((\widetilde N,\widetilde g)\) is of dimension 1 or 2 or a space of constant curvature.
The authors present examples of such space-times. Pseudosymmetric warped products of semi-Riemannian spaces of constant curvature are investigated. In particular, a curvature characterization of some class of Robertson-Walker type space-times is obtained. An example of a warped product of spaces of constant curvature which can be locally realized as a hypersurface in a space of constant curvature is presented.

MSC:
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53C35 Differential geometry of symmetric spaces
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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