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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.

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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