Semi-invariant warped product submanifolds of cosymplectic manifolds

Meraj Ali Khan,Rashmi Sachdeva,Siraj Uddin

doi:10.1186/1029-242x-2012-19

Abstract

In this article, we obtain the necessary and sufficient conditions that the semi-invariant submanifold to be a locally warped product submanifold of invariant and anti-invariant submanifolds of a cosymplectic manifold in terms of canonical structures T and F. The inequality and equality cases are also discussed for the squared norm of the second fundamental form in terms of the warping function.

Highlights

Bishop and O’Neill [1] introduced the notion of warped product manifolds in order to construct a large variety of manifolds of negative curvature
He studied warped product CR-submanifolds of the form M = M⊥ ×l MT such that M⊥ is a totally real submanifold and MT is a holomorphic submanifold of a Kaehler manifold Mand proved that warped product CR-submanifolds are CR-products
He considered the warped product CR-submanifolds in the form of M = MT ×l M⊥ which are known as CR-warped products where MT and M⊥ are holomorphic and totally real submanifolds of a Kaehler manifold M, respectively

Summary

Introduction

Bishop and O’Neill [1] introduced the notion of warped product manifolds in order to construct a large variety of manifolds of negative curvature. Proposition 3.1 [9]Let M be a semi-invariant submanifold of a cosymplectic manifold the anti-invariant distribution D⊥ is integrable. Proposition 3.2 The invariant distribution D on a semi-invariant submanifold of a cosymplectic manifold is integrable if and only if g(h(X, φY), φZ) = g(h(φX, Y), φZ)

Results

Conclusion