On holonomy groups of K-contact sub-pseudo-Riemannian manifolds

TL;DR

This paper studies the holonomy groups of K-contact sub-pseudo-Riemannian manifolds, proving their possible structures and classifying related Lorentzian holonomy algebras, with applications to sub-Lorentzian geometry.

Contribution

It establishes the relationship between horizontal and adapted holonomy groups in indefinite signature settings and classifies codimension-one ideals for Lorentzian holonomy algebras.

Findings

Horizontal holonomy group is either the adapted holonomy group or a codimension-one normal subgroup.

In sub-Lorentzian cases, the adapted holonomy group matches the holonomy of a Lorentzian manifold.

Complete classification of codimension-one ideals for Lorentzian holonomy algebras.

Abstract

This article investigates the holonomy groups of K-contact sub-pseudo-Riemannian manifolds. The primary result is a proof that the horizontal holonomy group either coincides with the adapted holonomy group or acts as its normal subgroup of codimension one. The theory is adapted for metrics of indefinite signature, bypassing the problem of subspace degeneracy that previously prevented the use of established orthogonal decomposition methods. It is established that, in the sub-Lorentzian case, the adapted holonomy group corresponds to the holonomy group of a certain Lorentzian manifold. This work also provides a complete classification of codimension-one ideals for Lorentzian holonomy algebras and presents specific examples of structures based on Cahen-Wallach spaces and K\"ahler manifolds.