Geometry and holonomy of indecomposable cones

Abstract

We study the geometry and holonomy of semi-Riemannian, time-like metric cones that are indecomposable, i.e., which do not admit a local decomposition into a semi-Riemannian product. This includes irreducible cones, for which the holonomy can be classified, as well as non irreducible cones. The latter admit a parallel distribution of null $k$-planes, and we study the cases $k=1$ and $k=2$ in detail. In these cases, i.e., when the cone admits a distribution of parallel null tangent lines or planes, we give structure theorems about the base manifold. Moreover, in the case $k=1$ and when the base manifold is Lorentzian, we derive a description of the cone holonomy. This result is obtained by a computation of certain cocycles of indecomposable subalgebras in $\mathfrak{so}(1,n-1)$.

42 pages; in v2 the proofs in Sections 5.2 and 5.3 are shortened by using results by Hochschild and Serre

References in corpus (3)

• On the local structure of Lorentzian Einstein manifolds with parallel distribution of null lines
• Gallot-Tanno theorem for pseudo-Riemannian metrics and a proof that decomposable cones over closed complete pseudo-Riemannian manifolds do not exist
• Holonomy groups and special geometric structures of pseudo-Kählerian manifolds of index 2

Cited by in corpus (1)

• Semi-Riemannian cones