Sergey Galaev

By Sergey Galaev and Evgeny Kokin

Quasi-semi-Weyl and quasi-statistical structures are based on a connection with torsion. In this chapter, 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, as well as with the help of an endomorphism that preserves the indicated 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 sub-Riemannian quasi-semi-Weyl and sub-Riemannian quasi-statistical structures are introduced, which are modifications of quasi-semi-Weyl and quasi-statistical structures for the case of sub-Riemannian manifolds of contact type. The structural endomorphism for the connection of a sub-Riemannian quasi-statistical structure is found. As an example, we consider non-holonomic 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 distribution of such manifold differs from the identity transformation only by a factor.

Part of the book: Topology