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

A bi-warped product of the form: M = N-T x f(1) N-perpendicular to(n1) x f(2) N-theta(n2) in a contact metric manifold is called a CRS bi-warped product, where N-T, N-perpendicular to(n1) and N-theta(n2) are invariant, anti-invariant and proper pointwise slant submanifolds, respectively. First, we prove that there are no proper CRS bi-warped products other than contact CR-biwarped products in any Sasakian manifold. Then, we prove that if M is a CRS bi-warped product in a cosymplectic manifold, its second fundamental form h satisfies parallel to h parallel to(2) >= 2n(1)parallel to del(1n f(1))parallel to(2) + 2n(2)(1 + 2 cot(2) theta)parallel to del(1n f(2))parallel to(2). Several applications of this inequality are given. Finally, we provide a non-trivial example of CRS bi-warped product which satisfies the equality case.

Automated summary

This paper investigates CRS bi-warped products in contact metric manifolds and proves that there are no proper CRS bi-warped products other than contact CR-biwarped products in Sasakian manifolds. It also provides an important inequality and several applications for CRS bi-warped products in cosymplectic manifolds.