Generic warped products in locally product Riemannian manifolds

In this paper, we introduce a new class of warped products, called generic warped product submanifolds in locally product Riemannian manifolds with pointwise slant fiber. We prove that every generic warped product submanifold B x(f) M-theta in a locally product Riemannian manifold satisfies the following inequality: parallel to h parallel to(2) >= s[cos(2) theta parallel to(del) over right arrow (perpendicular to)(Inf)parallel to(2) + 2(csc theta)(2) parallel to(del) over right arrow (perpendicular to)(Inf )parallel to(2)] where B = M-T x M-perpendicular to, a semi-invariant submanifold: M-theta is a pointwise slant submanifold of dimension s and (del) over right arrow (T)(Inf) and (del) over right arrow (perpendicular to)(Inf) are gradient components of the warping function Inf along M-T and M-perpendicular to, respectively. The equality case of the lower bound is also considered. Furthermore, we give many applications of this inequality and construct some non-trivial examples. (C) 2019 Elsevier B.V. All rights reserved.