Contact Cr-Warped Product Submanifolds of Nearly Trans-Sasakian Manifolds

Generalized inequalities on warped product submanifolds in nearly trans-Sasakian manifolds

“…In section 4, the necessary lemmas for the two inequalities and some geometric obstructions are obtained. In section 5, a general inequality which generalizes obtained inequalities in [16,18,19,24] is established. In section 6, we develop a new technique by means of Gauss equation and apply to construct a general inequality for the second fundamental form in terms of the scalar curvatures of submanifolds and the warping function.…”

“…We establish two general inequalities for the squared norm of the second fundamental form. The first inequality generalizes derived inequalities for some contact metric manifolds [16,18,19,24], while by a new technique, the second inequality is constructed to express the relation between extrinsic invariant (second fundamental form) and intrinsic invariant (scalar curvatures). The equality cases are also discussed.…”

The First Eigenvalue Estimates of Warped Product Pseudo-Slant Submanifolds

“…The work of Chen is about the characterization of CR-warped products in Kaehler manifolds, and derives the inequality for the second fundamental form. In fact, distinct classes of warped product submanifolds of the different kinds of structures were studied by several geometers (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14]). Recently, Ali et al [15], established general inequalities for warped product pseudo-slants isometrically immersed in nearly Kenmotsu manifolds for mixed, totally geodesic submanifolds.…”

“…For contradict that warped product pseudo-slant submanifolds always not generalize CR-warped product submanifold which was show in [13]. However, some interesting inequalities have been obtained by many geometers (see [4,10,12,[16][17][18][19][20]) for distinct warped product submanifolds in the different types of ambient manifolds. In [5], Al-Solamy derived the inequality for mixed, totally geodesic warped product pseudo-slant submanifolds of type M = M θ × f M ⊥ , in a nearly cosymplectic manifold.…”

On warped product semi-slant submanifolds of nearly Trans-Sasakian manifolds

“…A contact CR-warped product submanifold is the Riemannian product of invariant and anti-invariant submanifold. It was proved in [21] that there does not exist any contact CR-warped product of type M = N ⊥ × f N T , of nearly Trans-Sasakian manifolds in both cases when structure field tangent to either base manifold or fiber. Also, it was also found in the same paper, the non-trivial contact CR-warped product of the form M = N T × f N ⊥ , in a nearly Trans-Sasakian manifold such that N T invariant tangent to Reeb vector field.…”