Almost quasi-Sasakian manifolds equipped with skew-symmetric connection

TL;DR

This paper explores the properties of almost quasi-Sasakian manifolds with a focus on a special skew-symmetric connection, establishing conditions under which these manifolds are Einstein, and differentiating between even and odd rank structures.

Contribution

It introduces the concept of a canonical connection on almost quasi-Sasakian manifolds and identifies conditions for these manifolds to be Einstein, expanding understanding of their geometric structure.

Findings

The canonical connection is not metric in even rank cases.

Conditions are established for almost quasi-Sasakian manifolds to be Einstein.

The transversal structure is shown to be Kähler under certain conditions.

Abstract

On a sub-Riemannian manifold, a connection with skew-symmetric torsion is defined as the unique connection from the class of $N$-connections that has this property. Two cases are considered separately: sub-Riemannian structure of even rank, and sub-Riemannian structure of odd rank. The resulting connection, called the canonical connection, is not a metric connection in the case when the sub-Riemannian structure is of even rank. The structure of an almost quasi-Sasakian manifold is defined as an almost contact metric structure of odd rank that satisfies additional requirements. Namely, it is required that the canonical connection is a metric connection and that the transversal structure is a K\"ahler structure. Both the quasi-Sasakian structure and the more general almost contact metric structure, called an almost quasi-Sasakian structure, satisfy these requirements. Sufficient conditions are found for an almost quasi-Sasakian manifold to be an Einstein manifold.