Abstract
In 2008, Chen and Dillen obtained a sharp estimation for the squared norm of the second fundamental form of multiply warped CR-submanifold M = M-1 x(f2) M-2 x... x(fk) M-k in an arbitrary Kahler manifold M such that M-1 is a holomorphic submanifold and M-perpendicular to = f(2) M-2 x ... x(fk) M-k is a totally real submanifold of M. In this paper, we study bi-warped product submanifolds of locally product Riemannian manifolds which are the generalizations of single warped products. We prove that the bi-warped products of the form M-T x(f1) M-perpendicular to xf(2) M. and M-perpendicular to x(f1) MT x(f2) M. in an arbitrary locally product Riemannian manifold M, where MT is an invariant submanifold, M. an anti-invariant submanifold and M. a slant submanifold of M, are either Riemannian products or single warped products. Then, we investigate the geometry of bi-warped product submanifolds M-theta xf(1) MT xf(2) M. in a locally product Riemannian manifold M. We provide non-trivial examples of such submanifolds and a sharp estimation for the squared norm of the second fundamental form is obtained in terms of the warping functions f(1) and f(2). The equality case is also considered. Further, we give some applications of our main result.