Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 50, No. 2, pp. 337352 (2009) 

Generic warped product submanifolds in nearly Kaehler manifoldsViqar Azam Khan and Khalid Ali KhanDepartment of Mathematics, College of Science, P.O. Box 80203, King Abdul Aziz University, Jeddah21589, K.S.A., email: viqarster@gmail.com; School of Engineering and Logistics, Faculty of Technology, Charles Darwin University, NT0909, Australia, email: khalid.mathematics@gmail.comAbstract: Warped product manifolds provide excellent setting to model spacetime near black holes or bodies with large gravitational force (cf. [Be], [Bi], [H]). Recently, results are published exploring the existence (or nonexistence) of warped product submanifolds in Kaehlerian and contact settings (cf. [C1], [M], [S]). To continue the sequel, we have considered warped product submanifolds of nearly Kaehler manifolds with one of the factors a holomorphic submanifold. Such submanifolds are generic submanifolds in the sense of B. Y. Chen [C2] and provide a generalization of CR and semislant submanifolds. It is shown that nearly Kaehler manifolds do not admit nontrivial warped product generic submanifolds, thereby generalizing the results of Chen [C1] and Sahin [Sa]. However, nontrivial generic warped products (obtained by reversing the two factors of warped product generic submanifolds) exist in nearly Kaehler manifolds (cf. [Se21]). Some interesting results on the geometry of these submanifolds are obtained in the paper. [Be] Beem, J. K.; Ehrlich, P. E.; Easley, K.: \textit{Global Lorentzian geometry}. Marcel Dekker, New York 1996. [Bi] Bishop, R. L.; O'Neill, B.: \textit{Manifolds of Negative curvature}. Trans. Am. Math. Soc. {\bf 145} (1969), 149. [H] Hong, S. T.: Warped products and black holes. Nuovo Cim. J. B {\bf 120} (2005), 12271234. [C1] Chen, B.Y.: \textit{Geometry of warped product CRsubmanifolds in Kaehler Manifolds}. Monatsh. Math. {\bf 133} (2001), 177195. [M] Munteanu, M. I.: A note on doubly warped product contact CRsubmanifolds in transSasakian manifolds. Acta Math. Hung. {\bf 116}(12) (2007), 121126. [Sa] Sahin, B.: Non existence of warped product semislant submanifolds of Kaehler manifolds. Geom. Dedicata {\bf 117} (2006), 195202. [C2] Chen, B.Y.: \textit{Differential Geometry of Real Submanifolds in a Kaehler Manifold}. Monatsh. Math. {\bf 91} (1981), 257275. [Se] Sekigawa, K.: Some CRsubmanifolds in a $6$dimensional sphere. Tensor (New Ser.) {\bf 41} (1984), 1320. Keywords: nearly Kaehler manifold, warped product, slant submanifold, semislant submanifold, generic warped products Classification (MSC2000): 53C40, 53C42, 53C15 Full text of the article:
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