Pointwise slant and pointwise semi-slant submanifolds in almost contact metric manifolds

Abstract

As a generalization of slant submanifolds and semi-slant submanifolds, we introduce the notions of pointwise slant submanifolds and pointwise semi-slant sunmanifolds of an almost contact metric manifold. We obtain a characterization at each notion, investigate the topological properties of pointwise slant submanifolds, and give some examples of them. We also consider some distributions on cosymplectic, Sasakian, Kenmotsu manifolds and deal with some properties of warped product pointwise semi-slant submanifolds. Finally, we give some inequalities for the squared norm of the second fundamental form in terms of a warping function and a semi-slant function for warped product submanifolds of cosymplectic, Sasakian, Kenmotsu manifolds.

Warped Product Pointwise Semi-slant Submanifolds of Almost Contact Manifolds

“…The non-existence of warped product pointwise semi-slant submanifolds of the form M θ × f M T of Kaehler and Sasakian manifolds is proved in [26] and [23]. On the other hand, there exist a non-trivial warped product pointwise semi-slant submanifolds of the form M T × M θ of Kaehler manifolds [26] and contact metric manifolds [23].…”

“…In a similar way, K.S. Park [23] defined and studied pointwise slant submanifols of almost contact metric manifolds. His definition of pointwise slant submanifolds of almost contact metric manifold is similar to the pointwise slant submanifolds of almost Hermitian manifolds, therefore we have modified his definition by considering the structure vector field ξ is tangent to the submanifold and studied pointwise slant submanifolds of almost contact metric manifolds in [31].…”

“…They have obtained several fundamental results, in particular, a characterization of these submanifolds. Later, K. S. Park [23] has extended this study for almost contact metric manifolds. In his definition of pointwise slant submanifolds of almost contact metric manifolds he did not mention whether the structure vector field ξ is either tangent or normal to the submanifold.…”

“…Recently, K. S. Park [23] studied warped product pointwise semi-slant submanifolds of almost contact metric manifolds. He proved that there do not exist warped product pointwise semi-slant submanifolds of the form M θ × f M T in M , where M is either a cosymplectic manifold, a Sasakian manifold or a Kenmotsu manifold such that M θ and M T are proper pointwise slant and invariant submanifolds of M , respectively.…”

Pointwise Semi-Slant Warped Product Submanifold in a Lorentzian Paracosymplectic Manifold

“…Later on, Sahin [24] continued the study of pointwise slant submanifold by presenting a new class of submanifolds called warped product pointwise semi-slant submanifolds in Kählerian manifolds. Recently, Park [22,23] and Balgeshir [2] extended the notion of pointwise slant, pointwise semi-slant submanifolds and pointwise almost h-semi-slant submanifolds along with its warped products aspects in almost contact and quaternionic Hermitian settings. Motivated by the works of these, in this research we introduced the pointwise semi-slant submanifolds in Lorentzian almost paracontact manifolds which can be considered as the generalization of slant, pointwise slant, semi-invariant, semi-slant submanifolds and investigate the warped aspects for such submanifold.…”