$k$-slant distributions

Abstract

Inspired by the concepts of slant distribution and slant submanifold, with their variants of hemi-slant, semi-slant, bi-slant, or almost bi-slant, we introduce the more general concepts of $k$-slant distribution and $k$-slant submanifold in the settings of an almost Hermitian, an almost product Riemannian, an almost contact metric, and an almost paracontact metric manifold and study some of their properties. We prove that, for any proper $k$-slant distribution in the tangent bundle of a Riemannian manifold, there exists another one in its orthogonal complement, and we establish basic relations (metric properties, formulae relating the involved tensor fields, conformal properties) between them. Furthermore, allowing the slant angles to depend on the points of the manifold, we generalize these concepts and those of pointwise slant distribution and pointwise slant submanifold to the concepts of $k$-pointwise slant distribution and $k$-pointwise slant submanifold in the above-mentioned settings. For any $k$-pointwise slant distribution, we prove the existence of a corresponding one in its orthogonal complement and reveal basic relations between them. We also provide sufficient conditions for $k$-pointwise slant distributions to become $k$-slant distributions and establish other related results. By the end, for the fulfilment of some specific requirements, we introduce a special class of $k$-pointwise slant distributions, that of pointwise $k$-slant distributions, and the corresponding class of submanifolds, pointwise $k$-slant submanifolds, which is slightly more general than the class of generic submanifolds in sense of Ronsse, getting new results.