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Conformal holonomy of C-spaces, Ricci-flat, and Lorentzian manifolds

Thomas Leistner
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Abstract

The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebra of a $C$-space is a Berger algebra. For Ricci-flat spaces we show how the conformal holonomy can be obtained by the holonomy of the ambient metric and get results for Riemannian manifolds and plane waves.

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Thomas Leistner. 2014-11-12. Conformal holonomy of C-spaces, Ricci-flat, and Lorentzian manifolds. https://arxiv.org/abs/math/0501239

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