CR-Warped Submanifolds in Kaehler Manifolds

The warped product $$N_1\times _f N_2$$ N 1 × f N 2 of two Riemannian manifolds $$(N_1,g_1)$$ ( N 1 , g 1 ) and $$(N_2,g_2)$$ ( N 2 , g 2 ) is the product manifold $$N_1\times N_2$$ N 1 × N 2 equipped with the warped product metric $$g=g_1+f^2 g_2$$ g = g 1 + f 2 g 2 , where f is a positive function on $$N_1$$ N 1 . Warped products play very important roles in differential geometry as well as in physics. A submanifold M of a Kaehler manifold $$\tilde{M}$$ M ~ is called a CR-warped product if it is a warped product $$M_T\times _f N_\perp $$ M T × f N ⊥ of a complex submanifold $$M_T$$ M T and a totally real submanifold $$M_\perp $$ M ⊥ of $$\tilde{M}$$ M ~ . In this article we survey recent results on warped product and CR-warped product submanifolds in Kaehler manifolds. Several closely related results will also be presented.