A NOTE ON GCR-LIGHTLIKE WARPED PRODUCT SUBMANIFOLDS IN INDEFINITE KAEHLER MANIFOLDS
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We prove the non-existence of warped product GCR-lightlike submanifolds of the type K⊥ ×λ KT such that KT is a holomorphic submanifold and K⊥ is a totally real submanifold in an indefinite Kaehler manifold K˜ . Further, the existence of GCR-lightlike warped product submanifolds of the type KT ×λ K⊥ 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 ω. Moreover, we also derive a geometric estimate for the second fundamental form of GCRlightlike warped product submanifolds, in terms of the Hessian of the warping function λ.
Abstract
We prove the non-existence of warped product GCR-lightlike submanifolds of the type K⊥ ×λ KT such that KT is a holomorphic submanifold and K⊥ is a totally real submanifold in an indefinite Kaehler manifold K˜ . Further, the existence of GCR-lightlike warped product submanifolds of the type KT ×λ K⊥ 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 ω. Moreover, we also derive a geometric estimate for the second fundamental form of GCRlightlike warped product submanifolds, in terms of the Hessian of the warping function λ.
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