Geometry of sub-Riemannian manifolds equipped with a quasi-semi-Weyl structure

Abstract

The quasi-semi-Weyl and quasi-statistical structures are based on a connection with torsion. In this paper, as a connection with torsion, we consider the so-called extended connection, which is defined with the help of an intrinsic connection, i.e., a connection in the distribution of a sub-Riemannian manifold, and with the help of an endomorphism that preserves the specified distribution and is called a structural endomorphism. It is proved that the extended connection is a connection of the quasi-semi-Weyl structure of a sub-Riemannian manifold of contact type only if the distribution of the sub-Riemannian manifold is involutive. In order to be able to consider sub-Riemannian manifolds with a not necessarily involutive distribution, the concepts of a sub-Riemannian quasi-semi-Weyl structure and a sub-Riemannian quasi-statistical structure are introduced in the paper, which are a modifications to the case of sub-Riemannian manifolds of contact type of quasi-semi-Weyl and quasi- statistical structure. The form of a structural endomorphism for the connection of a sub-Riemannian quasi-statistical structure is found. As an example, we consider nonholonomic Kenmotsu manifolds, which are sub-Riemannian manifolds of contact type endowed with an additional structure. It is proved that the restriction of the structural endomorphism to the distributions of such manifolds differs from the identity transformation only by a factor. It turns out that the intrinsic connection underlying the extended connection of a sub-Riemannian quasi-semi-Weyl structure on a sub-Riemannian manifold of contact type is compatible with the metric constraint tensor on the distribution of this manifold.