D.V. Alekseevsky, V. Cortés, A.S. Galaev, T. Leistner
Cones over Pseudo-Riemannian Manifolds and their Holonomy
Preprint series: ESI preprints

MSC:

53C05 Connections, general theory

53C50 Lorentz manifolds, manifolds with indefinite metrics

Abstract: By a classical theorem of Gallot (1979), a Riemannian cone over a
complete Riemannian manifold is either flat or has irreducible holonomy.
We consider metric cones with reducible holonomy
over pseudo-Riemannian manifolds. First we describe
the local structure of the base of the cone
when the holonomy of the cone is decomposable.
For instance, we find that the holonomy algebra of the base is always
the full pseudo-orthogonal Lie algebra. One of the global results
is that a cone over a
compact and complete pseudo-Riemannian manifold is either flat or has
indecomposable holonomy. Then we analyse the case when the cone has
indecomposable but reducible holonomy, which means that it admits a
parallel isotropic distribution. This analysis is carried out, first in the case where the
cone admits two complementary distributions and, second for
Lorentzian cones. We show that
the first case occurs precisely when the local geometry of the
base manifold is para-Sasakian and that of the cone is para-K\"ahlerian.
For Lorentzian cones we
get a complete description of the possible (local) holonomy
algebras in terms of the metric of the base manifold.
Keywords: Holonomy groups, pseudo-Riemannian cones, doubly warped products, para-Sasakian and para-Kahler structures