A general inequality for warped product $CR$-submanifolds of Kähler manifolds

Abstract

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.

Geometry of Warped Product Hemi-Slant Submanifolds of an S-Manifold

“…for any Z, W ∈ TM ⊥ and X ∈ TM θ . Hence, by the relations (30), (31) and the fact that M ⊥ is totally umbilical in M [22], we find that M ⊥ is totally umbilical submanifold in M. This completes the proof.…”

“…He showed several fundamental results on the existence of CR-warped products in Kaehler manifolds, such as optimal inequalities and characterizatios in [24][25][26]. Many geometers researched warped product submanifolds for the various structures on Riemannian manifolds, as inspired by Chen's work [27][28][29][30][31][32]. Some researchers have also extended this approach to warped product semi-slant and pseudo-slant submanifolds (see [32][33][34][35][36]).…”

First Chen Inequality for General Warped Product Submanifolds of a Riemannian Space Form and Applications

“…For example, the relativistic model of the Schwarzschild spacetime that describes the outer space around a massive star or a black hole admits a warped product construction [1]. For this reason, in the years, many geometric aspects of the various types of warped product submanifods have been studied in various ambient spaces, (among others, see for example [2], [3], [4], [5], [6], [7], [8], [9]).…”