16 research outputs found
The Geometry of Warped Product Singularities
In this article the degenerate warped products of singular semi-Riemannian
manifolds are studied. They were used recently by the author to handle
singularities occurring in General Relativity, in black holes and at the
big-bang. One main result presented here 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 connection
and the Riemann curvature 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. Applications include
cosmological models and black holes solutions with semi-regular singularities.
Such singularities are compatible with a certain reformulation of the Einstein
equation, which in addition holds at semi-regular singularities too.Comment: 14 page
Realizations of Real Low-Dimensional Lie Algebras
Using a new powerful technique based on the notion of megaideal, we construct
a complete set of inequivalent realizations of real Lie algebras of dimension
no greater than four in vector fields on a space of an arbitrary (finite)
number of variables. Our classification amends and essentially generalizes
earlier works on the subject.
Known results on classification of low-dimensional real Lie algebras, their
automorphisms, differentiations, ideals, subalgebras and realizations are
reviewed.Comment: LaTeX2e, 39 pages. Essentially exetended version. Misprints in
Appendix are correcte