Cohomology of Hemi-slant Submanifolds of a Kaehler Manifold

Quasi hemi Slant Submanifolds of Lorentzian Concircular Structures

“…Let 𝑀 be a submanifold of a (𝐿𝐶𝑆) 𝑛 -manifold 𝑀 ~ with induced metric 𝑔. Also, let 𝛻 and 𝛻 ⊥ be the induced connections on 𝑇𝑀 and 𝑇 ⊥ 𝑀 of 𝑀 respectively, then Gauss and Weingarten formula are given by: 𝛻 ~𝑋𝑌 = 𝛻 𝑋 𝑌 + ℎ(𝑋, 𝑌) (21) 𝛻 ~𝑋𝑉 = −𝐴 𝑉 𝑋 + 𝛻 𝑋 ⊥ 𝑉 (22) for all 𝑋, 𝑌 ∈ 𝛤(𝑇𝑀) and 𝑉 ∈ 𝛤(𝑇 ⊥ 𝑀), where ℎ is second fundamental form and 𝐴 𝑉 is shape operator. These are related as follows:…”

“…[16], studied submanifolds of (𝐿𝐶𝑆) 𝑛 -manifolds. This concept was studied by many authors in differentiable manifolds, [17], [18], [19], [20], [21], [22]. In [11], the authors examine the geometry of hemi-slant 𝜉 ⊥ -Lorentzian submersions from (𝐿𝐶𝑆) 𝑛 -manifolds.…”

The de Rham Cohomology Group of a Hemi-Slant Submanifold in Metallic Riemannian Manifolds

“…S. Ianuş, S. Marchiafava and G. E. Vîlcu [24] demonstrated that there is a canonical de Rham cohomology group for a closed paraquaternionic CR submanifold of paraquaternionic Kähler manifolds and discussed necessary conditions for such a cohomology group to be non-trivial. In [33], F. Şahin found some necessary conditions for any hemi-slant submanifold of a Kaehler manifold to define a non-trivial de Rham cohomology group. For a semi-invariant submanifold in locally product Riemannian manifolds, G. Pitis [29] obtained a close relation between its de Rham cohomology group and associated distributions; moreover, the author gave some examples which substantiate his claims.…”

A note on quasi-bi-slant submanifolds of Sasakian manifolds

“…Primarily, the hemi-slant submanifolds were known as anti-slant submanifolds. Later, Sahin [21] named these submanifolds as hemi-slant submanifolds. Hemi-slant submanifolds are one of the classes of bi-slant submanifolds.…”