# Geometry and holonomy of indecomposable cones

## TL;DR

This paper investigates the geometry and holonomy properties of indecomposable semi-Riemannian time-like cones, providing classifications and structure theorems for cases with parallel null distributions, especially in Lorentzian settings.

## Contribution

It offers new classifications and structure theorems for indecomposable semi-Riemannian cones with parallel null distributions, including holonomy descriptions in Lorentzian cases.

## Key findings

- Classification of holonomy for irreducible cones
- Structure theorems for cones with null distributions
- Holonomy description for Lorentzian base manifolds

## 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 $\mathfrak{so}(1,n-1)$.