Prolonged almost quazi-Sasakian structures

Abstract

The notion of an almost quasi-Sasakian manifold is introduced. A ma­nifold with an almost quasi-Sasakian structure is a generalization of a quasi-Sasakian manifold; the difference is that an almost quasi-Sasakian manifold is almost normal. A characteristic criterion for an almost quasi-Sasakian manifold is formulated. Conditions are found under which al­most quasi-Sasakian manifolds are quasi-Sasakian manifolds. In particu­lar, an almost quasi-Sasakian manifold is a quasi-Sasakian manifold if and only if the first and second structure endomorphisms commute. An extended almost contact metric structure is defined on the distribution of an almost contact metric manifold. It follows from the definition of an extended structure that it is a quasi-Sasakian structure only if the original structure is cosymplectic with zero Schouten curvature tensor. It is proved that the constructed extended almost contact metric structure is the struc­ture of an almost quasi-Sasakian manifold if and only if the Schouten ten­sor of the original manifold is equal to zero. Relationships are found be­tween the second structure endomorphisms of the original and extended structures.