Warped Products of Singular Semi-Riemannian Manifolds

Details Warped Products of Singular Semi-Riemannian Manifolds Warped Products of Singular Semi-Riemannian Manifolds

2011-05-17

arxiv.org/abs/1105.3404v3

Mathematics

Differential Geometry

Scientific paper

This article studies the degenerate warped products of singular semi-Riemannian manifolds. One main result is that a degenerate warped product of semi-regular semi-Riemannian manifolds with the warping function satisfying a certain condition is a semi-regular semi-Riemannian manifold. The main invariants of the warped product are expressed in terms of those of the factor manifolds. Examples of singular semi-Riemannian manifolds which are semi-regular are constructed as warped products. Degenerate warped products are used to define spherical warped products. As applications, cosmological models and black holes solutions with semi-regular singularities are constructed. Such singularities are compatible with the densitized version of Einstein's equation, and don't block the time evolution. In following papers we will apply the technique developed here to resolve the singularities of the Friedmann-Lemaitre-Robertson-Walker, Schwarzschild, Reissner-Nordstrom and Kerr-Newman spacetimes.

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