Lightlike foliations on Lorentzian manifolds with weakly irreducible holonomy algebra

TL;DR

This paper investigates the structure of lightlike foliations on Lorentzian manifolds with specific holonomy algebras, providing new global equations, decomposition of curvature, and criteria for classifying holonomy types.

Contribution

It introduces generalized structure equations for lightlike foliations and a method to classify holonomy algebra types using global operators on Lorentzian manifolds.

Findings

Global structure equations for lightlike foliations

Decomposition of the curvature tensor into components

Criteria for identifying holonomy algebra types

Abstract

We study the lightlike foliations that appear on Lorentzian manifolds with weakly irreducible not irreducible holonomy algebra. We give global structure equations for the foliation that generalize the Gauss and Weingarten equations for one lightlike hypersurface. This gives us some global operators on the manifold. Using these operators, we decompose the curvature tensor of the manifold into several components. We give a criteria how to find the type of the holonomy algebras (there are 4 possible types) in terms of the global operators.