О почти квазисасакиевых параконтактных структурах на распределениях субримановых многообразий

Abstract

The concept of an almost quasi-Sasakian paracontact structure is introduced. In contrast to the quasi-Sasakian paracontact structure, the introduced structure does not have the normality property. The invariants of the intrinsic geometry of an almost quasi-Sasakian paracontact structure are studied. The intrinsic geometry of an almost paracontact metric manifold is understood to mean those of its geometric properties that depend only on the parallel translation determined by the intrinsic connection. In particular, it is proved that the vanishing of the Schouten tensor implies that the structure vector field is a Killing vecto r field. An example of an almost quasi-Sasakian paracontact structure is given. Almost quasi-Sasakian paracontact structures arise naturally on distributions of sub-Riemannian manifolds. On a non-involutive distribution D of a sub-Riemannian manifold M of contact type, an almost paracontact metric structure is defined, which is called an extended structure. It is proved that on a distribution of zero curvature, the extended structure is an almost quasi -Sasakian paracontact structure. A characteristic property of a distribution of zero curvature is the vanishing of the Schouten curvature tensor. The well-known classification of extended structures is analyzed based on the properties of the fundamental tensor F of type (0, 3) associated to the extended structure. In accordance with this classification, there are 212 classes of almost paracontact metric structure, among which there are 12 basic ones. A class containing an extended almost quasi-Sasakian paracontact structure is found.