Warped product contact CR-submanifolds of globally framed f-manifolds with Lorentz metric

Abstract

In the present paper, we study globally framed f-manifolds in the particular setting of indefinite S-manifolds for both spacelike and timelike cases. We prove that if $M=N^{\perp }{\times }_f{N}^T$ is a warped CR-submanifold such that $N^{\perp }$ is $\varphi$?-anti-invariant and NT is $\varphi$?-invariant, then M is a CR-product. We show that the second fundamental form of a contact CR warped product of a indefinite S space form satisfies a geometric inequality, $\Vert h{\Vert }^2\ge p3\Vert \nabla lnf{\Vert }^2-△lnf+(c+2)k+1$.

arXiv admin note: text overlap with arXiv:1106.6317