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

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.