Screen Generic Lightlike Submanifolds of Indefinite Sasakian Manifolds
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Screen Generic Lightlike Submanifolds of Indefinite Sasakian Manifolds Ram Shankar Gupta Abstract. In this paper, we introduce the general notion of screen generic lightlike submanifolds of indefinite Sasakian manifolds. We study screen generic lightlike submnaifolds, contact totally umbilical screen generic lightlike submanifolds, and minimal screen generic lightlike submanifolds of indefinite Sasakian manifolds. Also, we provide some examples of screen generic lightlike submanifold and minimal screen generic lightlike submanifold of an indefinite Sasakian manifold. Mathematics Subject Classification. 53C15, 53C40, 53C50, 53D15. Keywords. Degenerate metric, Almost contact manifold, Indefinite Sasakian manifold, Generic lightlike submanifold, Minimal lightlike submanifold.
1. Introduction In the theory of submanifolds of semi-Riemannian manifolds it is interesting to study the geometry of lightlike submanifolds as the intersection of normal vector bundle and the tangent bundle is non-trivial. The geometry of lightlike submanifold is used in mathematical physics, in particular, in general Relativity since lightlike submanifolds can be models of different types of horizons(event horizons, Cauchy’s horizons, Kruskal’s horizons). The geometry of lightlike submanifolds were introduced and presented in a book by Duggal and Bejancu [7]. They introduced a non-degenerate screen distribution to construct a non intersecting lightlike transversal vector bundle of the tangent bundle. In 2010, Duggal and Sahin published another book [11] on differential geometry of lightlike submanifolds. In this book they focused on all new geometric results on lightlike geometry with proofs and their physical applications in mathematical physics. The geometry of lightlike hypersurfaces and submanifolds were studied in [2,5–7,9–16,18–23,26] and references therein. This work is supported by award of grant under FRGS for the year 2019–20, F.No. GGSIPU/DRC/FRGS/2019/1553/5.
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A generic submanifold in a Kaehler manifold and that in a Sasakian manifold was mainly defined as the generalization of CR-submanifolds [4,25]. Since CR-submanifolds include holomorphic submanifolds and totally real submanifolds as subspaces, therefore the generic submanifolds are the most general class of submanifolds. Since invariant lightlike submanifolds and antiinvariant lighlike submanifolds of indefinite Hermitian manifolds are particular cases of screen Cauchy-Riemann lightlike submanifolds, therefore, generic lightlike submanifolds in lightlike geometry should include screen CauchyRiemann lightlike submanifolds. Generic lightlike submanifolds defined in [17] do not contain proper screen Cauchy-Riemann lightlike submanifolds. Therefore, the concept of screen generic lightlike submanifolds was introduced and studied in [5]. On the other hand in contact geometry, Duggal and Jin have studied the generic lightlike submanifolds of an indefinite Sasakian manifold and obtained some interesting results in [8]. However, a gen
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