A GENERAL INEQUALITY FOR WARPED PRODUCT CR-SUBMANIFOLDS OF KAHLER MANIFOLDS

In this paper, warped product CRCR-submanifolds in Kahler manifolds and contact Sasakian, Kenmotsu cosymplectic manifolds, are shown to possess a geometric property; namely DTDT-minimal. Taking benefit from property, an optimal general inequality is established by means of the Gauss equation, we leave cosyplectic because it easy structure. Moreover, rich geometry appears when necessity sufficiency proved discussed equality case. Applying inequality, inequalities obtained Munteanu derived as particular cases. Up now, method used Chen can not extended for ambient many limitations using Codazzi equation. Hence, Our depends on The constructed involve intrinsic invariant (scalar curvature) controlled extrinsic one (the second fundamental form), which provides answer well-know Chen's research problem (Problem 1.1???). As further directions, have addressed couple open problems arose naturally during work depending its results.