Resumen de Geometry of pointwise CR-slant warped products in Kaehler manifolds

Bang Yen Chen, Siraj Uddin, Falleh R. Al Solamy

• We call a submanifold M of a Kaehler manifold M˜ a pointwise CR-slant warped product if it is a warped product, B ×f Nθ, of a CR-product B = NT × N⊥ and a proper pointwise slant submanifold Nθ with slant function θ, where NT and N⊥ are complex and totally real submanifolds of M˜ . We prove that if a pointwise CR-slant warped product B×f Nθ with B = NT ×N⊥ in a Kaehler manifold is weakly Dθ -totally geodesic, then it satisfies kσk 2 ≥ 4s (csc2θ + cot2θ)k∇T(ln f)k2 + (cot2θ)k∇⊥(ln f)k2, where NT , N⊥, and Nθ are complex, totally real and proper pointwise slant submanifolds of M˜ , respectively, and s = 1/ 2 dim Nθ. In this paper we also investigate the equality case of the inequality. Moreover, we give a non-trivial example and provide some applications of this inequality.