On the geometry of generalized nonholonomic Kenmotsu manifolds

Abstract

The concept of a generalized nonholonomic Kenmotsu manifold is introduced. In contrast to the previously defined nonholonomic Kenmotsu manifold, the manifold studied in the article is an almost normal almost contact metric manifold of odd rank. The manifold is equipped with a metric connection with torsion, which is called the canonical connection in this work. The main properties of the canonical connection are studied. The canonical connection is an analogue of the generalized Tanaka-Webster connection. In this paper, we prove that the canonical connection is the only metric connection with torsion of a special structure that pre­serves the structural 1-form and the Reeb vector field. We study the in­trinsic geometry of a generalized nonholonomic Kenmotsu manifold equipped with a canonical connection. It is proved that if a generalized nonholonomic Kenmotsu manifold is an Einstein manifold with respect to a canonical connection, then it is Ricci-flat with respect to this connec­tion. An example of a generalized nonholonomic Kenmotsu manifold that is not a nonholonomic Kenmotsu manifold is given.

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

“…A normal almost contact metric manifold M is called a non-holonomic Kenmotsu manifold if L ξ ! g ¼ 2 g À η ⊗ η ðÞ holds true [19].…”

“…The Kenmotsu manifolds were discovered in 1972 in [18]. The concept of a non-holonomic Kenmotsu manifold was introduced in [19]. In [20], a non-holonomic Kenmotsu manifold is equipped with an N-connection ∇ N with torsion of a special type.…”