Generic warped products in locally product Riemannian manifolds

Abstract

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×fMθ in a locally product Riemannian manifold satisfies the following inequality: ‖h‖2≥s[cos2θ‖∇→⊥(lnf)‖2+2cscθ+cotθ2‖∇→T(lnf)‖2]where B=MT×M⊥, a semi-invariant submanifold; Mθ is a pointwise slant submanifold of dimension s and ∇→T(lnf) and ∇→⊥(lnf) are gradient components of the warping function lnf along MT and M⊥, 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.