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

Abstract

In this paper, warped product $CR$-submanifolds in Kahler manifolds and warped product contact $CR$-submanifolds in Sasakian, Kenmotsu and cosymplectic manifolds, are shown to possess a geometric property; namely $D_T$-minimal. Taking benefit from this property, an optimal general inequality is established 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.

Keywords

• Warped product CR-submanifolds
• mean curvature vector
• scalar curvature
• minimal submanifolds
• K¨ahler manifolds
• Gauss equation

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