Communications of the
Korean Mathematical Society CKMS

We prove the non-existence of warped product $GCR$-lightlike submanifolds of the type $K_{\bot} \times_{\lambda} K_{T}$ such that $K_{T}$ is a holomorphic submanifold and $K_{\bot}$ is a totally real submanifold in an indefinite Kaehler manifold $\tilde{K}$. Further, the existence of $GCR$-lightlike warped product submanifolds of the type $K_{T} \times_{\lambda} K_{\bot}$ is obtained by establishing a characterization theorem in terms of the shape operator and the warping function in an indefinite Kaehler manifold. Consequently, we find some necessary and sufficient conditions for an isometrically immersed $GCR$-lightlike submanifold in an indefinite Kaehler manifold to be a $GCR$-lightlike warped product, in terms of the canonical structures $f$ and $\omega$. Moreover, we also derive a geometric estimate for the second fundamental form of $GCR$-lightlike warped product submanifolds, in terms of the Hessian of the warping function $\lambda$.

Keywords: Indefinite Kaehler manifolds, $GCR$-lightlike submanifolds, warped product lightlike submanifolds

MSC numbers: Primary 53C50; Secondary 53C15, 53C40