A Note on $GCR$-Lightlike Warped Product Submanifolds in Indefinite Kaehler Manifolds
Commun. Korean Math. Soc.
Published online July 7, 2021
Sangeet Kumar and Megha Pruthi
SGTB Khalsa College, Sri Anandpur Sahib; SGTB Khalsa College, Sri Anandpur Sahib
Abstract : 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, 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 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 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 Hessian of warping function $\lambda$.
Keywords : indefinite Kaehler manifolds, $GCR$-lightlike submanifolds, warped product lightlike submanifolds
MSC numbers : 53C15, 53C40, 53C50
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