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Parabolicity and rigidity of spacelike hypersurfaces immersed in a Lorentzian Killing warped product

Eudes L. de Lima, Henrique F. de Lima, Eraldo A. Jr. Lima, Adriano A. Medeiros (2017)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we extend a technique due to Romero et al. establishing sufficient conditions to guarantee the parabolicity of complete spacelike hypersurfaces immersed into a Lorentzian Killing warped product whose Riemannian base has parabolic universal Riemannian covering. As applications, we obtain rigidity results concerning these hypersurfaces. A particular study of entire Killing graphs is also made.

Projective relatedness and conformal flatness

Graham Hall (2012)

Open Mathematics

This paper discusses the connection between projective relatedness and conformal flatness for 4-dimensional manifolds admitting a metric of signature (+,+,+,+) or (+,+,+,−). It is shown that if one of the manifolds is conformally flat and not of the most general holonomy type for that signature then, in general, the connections of the manifolds involved are the same and the second manifold is also conformally flat. Counterexamples are provided which place limitations on the potential strengthening...

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