In this paper, we study warped product submanifolds of nearly trans-Sasakian manifolds. The non-existence of the warped product semi-slant submanifolds of the type N θ × f NT is shown, whereas some characterization and new geometric obstructions are obtained for the warped products of the type NT × f N θ . 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.
We study warped product pseudo-slant submanifolds of a nearly cosymplectic manifold. We obtain some characterization results on the existence or nonexistence of warped product pseudoslant submanifolds of a nearly cosymplectic manifold in terms of the canonical structures P and F.
We study of warped product submanifolds, especially warped product hemi-slant submanifolds of LP-Sasakian manifolds. We obtain the results on the nonexistance or existence of warped product hemi-slant submanifolds and give some examples of LP-Sasakian manifolds. The existence of warped product hemi-slant submanifolds of an LP-Sasakian manifold is also ensured by an interesting example.
This article has three recurrent goals. Firstly, we prove the existence of a wide class of warped product submanifolds possessing a geometrical property; namely, D i -minimal warped product submanifolds. Secondly, the first Chen inequality is derived for this class of warped products in Riemannian space forms, this inequality involves intrinsic invariants (δ-invariant and sectional curvature) controlled by an extrinsic one (the mean curvature vector), which provides an answer for Problem 1. Thirdly, this inequality is applied to derive a necessary condition for the immersed submanifold to be minimal in Riemannian space forms, which presents a partial answer for the well-known problem proposed by S.S. Chern, Problem 2. Also, a concrete example is constructed to insure the existence of D i -minimal warped product submanifolds for warped products other than CR and contact CR-warped product submanifolds. For further research directions, we address a couple of open problems; namely Problem 3 and Problem 4.
In this paper, warped product contact CR-submanifolds in Sasakian, Kenmotsu and cosymplectic manifolds are shown to possess a geometric property; namely D_T-minimal. Taking benet from this property, an optimal general inequality for warped product contact CR-submanifolds is established in both Sasakian and Kenmotsu manifolds by means of the Gauss equation, we leave cosyplectic because it is an easy structure. Moreover, a rich geometry appears when the necessity and sufficiency are proved and discussed in the equality case.
Applying this general inequality, the inequalities obtained by Munteanu are derived as particular cases.
Up to now, the method used by Chen and Munteanu can not extended for general ambient manifolds, this is because many limitations in using Codazzi equation. Hence, Our method depends on the Gauss equation. The inequality is constructed to involve an intrinsic invariant (scalar curvature) controlled by an extrinsic one (the second fundamental form), which provides an answer for the well-know Chen’s research problem (Problem 1.1). As further research directions, we have addressed a couple of open problems arose naturally during this work and depending on its results.
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