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

Siraj Uddin, Adela Mihai, Ion Mihai, Awatif Al-Jedani

• In 2008, Chen and Dillen obtained a sharp estimation for the squared norm of the second fundamental form of multiply warped CR-submanifold M = M1 × f2 M2 × ... × fk Mk in an arbitrary Kähler manifold M˜ such that M1 is a holomorphic submanifold and M⊥ = f2 M2 × ··· × fk Mk 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 MT × f1 M⊥ × f2 Mθ and M⊥ × f1 MT × f2 Mθ in an arbitrary locally product Riemannian manifold M˜ , where MT is an invariant submanifold, M⊥ an anti-invariant submanifold and Mθ a slant submanifold of M˜ , are either Riemannian products or single warped products.

  Then, we investigate the geometry of bi-warped product submanifolds Mθ × f1 MT × f2 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 f1 and f2. The equality case is also considered.

  Further, we give some applications of our main result.