Semi-Riemannian cones with parallel null planes
Let \(\epsilon=\pm 1\). The metric \(\epsilon\)-cone over a pseudo-Riemannian manifold \((M,g)\) is the manifold \(\hat M=\mathbb{R}_+\times M\) with the metric \(\hat g=\epsilon (dt)^2+t^2g\). The cone is space-like if \(\epsilon=1\) and it is time-like if \(\epsilon=-1\). By the result of \textit{S. Gallot} [Ann. Sci. Éc. Norm. Supér. (4) 12, 235--267 (1978; Zbl 0412.58009)] the space-like cone over a complete Riemannian manifold is either flat or its holonomy group is irreducible. \textit{C. Bär} [Commun. Math. Phys. 154, No. 3, 509--521 (1993; Zbl 0778.53037)] showed that Killing spinors on a (pseudo-)Riemannian manifolds correspond to parallel spinors on the cone. The above two results and the work by \textit{M. Y. Wang} [Ann. Global Anal. Geom. 7, No. 1, 59--68 (1989; Zbl 0688.53007)] lead to the classifications of complete simply connected Riemannian manifolds admitting real Killing spinors. This motivates the study of the holonomy groups of the cones over pseudo-Riemannian manifolds. The first detailed investigation of these holonomy groups is given in [\textit{D. V. Alekseevsky} et al., J. Reine Angew. Math. 635, 23--69 (2009; Zbl 1181.53057)] where it was shown that the cone over a complete pseudo-Riemannian manifold may have reducible holonomy group. Moreover it is possible that the holonomy group preserves an isotropic subspace of the tangent space. The present paper provides a complete local description of pseudo-Riemannian manifolds \((M,g)\) such that the holonomy group of the time-like cone over \((M,g)\) preserves a two-dimensional totally isotropic subspace. For the entire collection see [Zbl 1497.53003].
scientific article; zbMATH DE number 7632936
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