On one class of holonomy groups in pseudo-Riemannian geometry

TL;DR

This paper introduces a new class of holonomy groups in pseudo-Riemannian geometry, showing they can be realized as centralizers of symmetric operators within special orthogonal groups, expanding understanding of geometric structures.

Contribution

It proves that the identity component of the centralizer of a g-symmetric operator in SO(g) can serve as a holonomy group for some Levi-Civita connection, revealing a novel class of holonomy groups.

Findings

Identifies a new class of holonomy groups in pseudo-Riemannian geometry.

Shows these groups are centralizers of symmetric operators in SO(g).

Provides a construction for realizing these groups as holonomy groups.

Abstract

We describe a new class of holonomy groups on pseudo-Riemannian manifolds. Namely, we prove the following theorem. Let g be a nondegenerate bilinear form on a vector space V, and L:V -> V a g-symmetric operator. Then the identity component of the centraliser of L in SO(g) is a holonomy group for a suitable Levi-Civita connection.