Geometry of $CRS$ bi-warped product submanifolds in Sasakian and cosymplectic manifolds

Abstract

In this paper, we study bi-warped product submanifolds of Sasakian manifolds, which are the natural generalizations of single warped products and Remannian products. We show that if M is a bi-warped product submanifold of the formwhere N T , N ⊥ and N θ are invariant, anti-invariant and proper pointwise slant submanifolds of M , respectively then the second fundamental form of M satisfies a general inequality:and h is the second fundamental form and ∇(ln f 1 ) and ∇(ln f 2 ) are the gradient components along N ⊥ and N θ , respectively. Some applications of this inequality are given and we provide some non-trivial examples of bi-warped products.