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 $\phi$?-anti-invariant and NT is $\phi$?-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, $|h|^2 \geq p{3|\nabla lnf|^2 - \triangle lnf + (c + 2)k + 1}$.