Geometry of bi-warped product submanifolds of locally product Riemannian manifolds

In 2008, Chen and Dillen obtained a sharp estimation for the squared norm of the second fundamental form of multiply warped CR-submanifold $$M=M_1\times _{f_2}M_2\times \ldots \times _{f_k}M_k$$ in an arbitrary Kahler manifold $${\tilde{M}}$$ such that $$M_1$$ is a holomorphic submanifold and $$M_\perp =_{f_2}M_2\times \cdots \times _{f_k}M_k$$ is a totally real submanifold of $${\tilde{M}}$$. 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 $$M_T\times _{f_1}M_\perp \times _{f_2}M_\theta $$ and $$M_\perp \times _{f_1}M_T\times _{f_2}M_\theta $$ in an arbitrary locally product Riemannian manifold $${\tilde{M}}$$, where $$M_T$$ is an invariant submanifold, $$M_\perp $$ an anti-invariant submanifold and $$M_\theta $$ a slant submanifold of $${\tilde{M}}$$, are either Riemannian products or single warped products. Then, we investigate the geometry of bi-warped product submanifolds $$M_\theta \times _{f_1}M_T\times _{f_2}M_\perp $$ in a locally product Riemannian manifold $${\tilde{M}}$$. 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.