On holonomy groups of K-contact sub-pseudo-Riemannian manifolds
E. A. Kokin
TL;DR
The paper extends holonomy theory to $K$-contact sub-pseudo-Riemannian manifolds with indefinite metrics, establishing a dichotomy: the horizontal holonomy $\operatorname{Hol}_x(\nabla^g)$ either coincides with the adapted holonomy $\operatorname{Hol}_x(\nabla^{\boldsymbol{\tau}})$ or forms a codimension-one normal subgroup of it. By leveraging the Wagner connection and handling potential degeneracy in the subspace, the authors show the horizontal holonomy aligns with the Lorentzian holonomy in the sub-Lorentzian case and classify codimension-one ideals for Lorentzian holonomy algebras, providing concrete Cahen–Wallach and Kähler-based examples. The results build on prior work by removing completeness assumptions and adapting methods to the sub-pseudo-Riemannian setting, enriching the structural understanding of holonomy in non-Riemannian geometry. Overall, the work offers a clear framework for relating horizontal, adapted, and Lorentzian holonomies and supplies explicit classifications and constructions to illuminate the possible holonomy configurations in these geometric contexts.
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 Kähler manifolds.