Isomorphism for the Holonomy Group of a K-Contact Sub-Riemannian Space

TL;DR

This paper establishes an isomorphism between the holonomy group of the adapted connection on a K-contact manifold and the Levi-Civita holonomy on its orbit space, providing a new proof of a de Rham-type theorem for these manifolds.

Contribution

It proves the holonomy group of the adapted connection is isomorphic to that of the Levi-Civita connection on the orbit space, linking sub-Riemannian and Riemannian geometries.

Findings

Holonomy group of adapted connection is isomorphic to that of Levi-Civita connection on the orbit space.

If the holonomy is not irreducible, the orbit space locally splits as a product of Riemannian manifolds.

Provides a simplified proof of the de Rham theorem for K-contact sub-Riemannian manifolds.

Abstract

The holonomy group of the adapted connection on a K-contact Riemannian manifold $(M, \theta, g)$ is considered. It is proved that if the orbit space $M/\xi$ of the Reeb field $\xi$ action admits a manifold structure, then the holonomy group of the adapted connection on $M$ is isomorphic to the holonomy group of the Levi-Civita connection on the Riemannian manifold $(M/\xi, h)$, where $h$ is the induced Riemannian metric on $M/\xi$. Thanks to this result, a simple proof of the de Rham theorem for the case of K-contact sub-Riemannian manifolds is obtained, stating that if the holonomy group of the adapted connection on $M$ is not irreducible, then the orbit space $M/\xi$ is locally a product of Riemannian manifolds.