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Radical anti-invariant lightlike submanifolds of semi-Riemannian product manifolds

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We introduce radical anti-invariant lightlike submanifolds of a semi Riemannian product manifold and give examples. After we obtain the conditions of integrability of distributions which are involved in the definition of radical anti-invariant lightlike submanifolds, we investigate the geometry of leaves of distributions. | Turk J Math 32 (2008) , 429 – 449. ¨ ITAK ˙ c TUB Radical Anti-Invariant Lightlike Submanifolds of Semi-Riemannian Product Manifolds Erol Kılı¸c and Bayram S ¸ ahin Abstract We introduce radical anti-invariant lightlike submanifolds of a semi Riemannian product manifold and give examples. After we obtain the conditions of integrability of distributions which are involved in the definition of radical anti-invariant lightlike submanifolds, we investigate the geometry of leaves of distributions. We also obtain the induced connection is a metric connection and a radical anti-invariant lightlike submanifold is a product manifold under certain conditions. Finally, we study totally umbilical radical anti-invariant lightlike submanifolds and observe that they are totally geodesic under a condition. Key Words: Degenerate Metric, Semi-Riemannian Product Manifold, r-Lightlike Submanifold, Locally Riemannian Product 1. Introduction The geometry of lightlike submanifolds of a semi-Riemannian manifold was presented in [5] (see also in [6]) by K. L. Duggal and A. Bejancu. In [5], they also introduced CR-lightlike submanifolds of indefinite Kaehler as lightlike version of non-degenerate CRsubmanifolds. But, they showed that such lightlike submanifolds do not contain invariant and anti-invariant submanifolds contrary to the non-degenerate CR-submanifolds. Therefore, in [8] (see also [9]), K. L. Duggal and B. Sahin introduced screen CR-lightlike submanifolds, and showed that such lightlike submanifolds include invariant lightlike 2000 AMS Mathematics Subject Classification: 53C15, 53C40, 53C42, 53C50. 429 ˙ KILIC ¸, S ¸ AHIN submanifolds as well as anti-invariant (screen real) submanifolds of indefinite Kaehler manifolds. They also showed that there are no inclusion relation between CR-lightlike submanifolds and SCR-lightlike submanifolds. Therefore, K. L. Duggal and B. Sahin introduced generalized CR-lightlike submanifolds of indefinite Kaehler manifolds as a generalization .