Semi-invariant warped product submanifolds of almost contact manifolds

Falleh R Al-Solamy,Meraj Ali Khan

doi:10.1186/1029-242x-2012-127

Falleh R Al-Solamy, Meraj Ali Khan

Open Access

https://doi.org/10.1186/1029-242x-2012-127

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Abstract

In this article, we have obtained necessary and sufficient conditions in terms of canonical structure F on a semi-invariant submanifold of an almost contact manifold under which the submanifold reduced to semi-invariant warped product submanifold. Moreover, we have proved an inequality for squared norm of second fundamental form and finally, an estimate for the second fundamental form of a semi-invariant warped product submanifold in a generalized Sasakian space form is obtained, which extend the results of Chen, Al-Luhaibi et al., and Hesigawa and Mihai in a more general setting. 2000 Mathematics Subject Classification: 53C25; 53C40; 53C42; 53D15.

Highlights

Bishop and O’Neil [1] introduced the notion of warped product manifolds
With regard to physical applications of these manifolds, one may realize that space time around a massive star or a black hole can be modeled on a warped product manifolds for instance and warped product manifolds are widely used in differential geometry, Physics and as well as in different branches of Engineering
Hesigawa and Mihai [9] obtained the inequality for squared norm of the second fundamental form in term of the warping function for contact CR-warped product in Sasakian manifolds

Summary

Introduction

Bishop and O’Neil [1] introduced the notion of warped product manifolds. These manifolds are generalization of Riemannian product manifolds and occur naturally, e.g., surface of revolution is a warped product manifold. Atceken [4] studied contact CR-warped product submanifolds in Cosymplectic space-forms and obtained an inequality for second fundamental form in terms of warping function. Since generalized Sasakian space forms include all the classes of almost contact metric manifold, so we have obtained an inequality for squared norm of second fundamental form for semiinvariant warped product submanifolds in the setting of generalized Sasakian space form. Given an almost contact metric manifold M (φ, ξ , η, g) , we say that Mis a generalized Sasakian space form if there exist three functions f1, f2, and f3 on Msuch that, the curvature tensor R is given by. For a warped product manifold N1 ×f N2, we denote by D1 and D2 the distributions defined by the vectors tangent to the leaves and fibers, respectively.

Xf f

On the other hand by Coddazi equation