Generic warped product submanifolds in nearly Kaehler manifolds

Viqar Azam Khan and Khalid Ali Khan

Abstract: Warped product manifolds provide excellent setting to model space-time near black holes or bodies with large gravitational force (cf. [Be], [Bi], [H]). Recently, results are published exploring the existence (or non-existence) 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 sub-manifolds are generic submanifolds in the sense of B. Y. Chen [C2] and provide a generalization of CR and semi-slant submanifolds. It is shown that nearly Kaehler manifolds do not admit non-trivial warped product generic submanifolds, thereby generalizing the results of Chen [C1] and Sahin [Sa]. However, non-trivial 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), 1--49. [H] Hong, S. T.: Warped products and black holes. Nuovo Cim. J. B {\bf 120} (2005), 1227--1234. [C1] Chen, B.-Y.: \textit{Geometry of warped product CR-submanifolds in Kaehler Manifolds}. Monatsh. Math. {\bf 133} (2001), 177--195. [M] Munteanu, M. I.: A note on doubly warped product contact CR-submanifolds in trans-Sasakian manifolds. Acta Math. Hung. {\bf 116}(1--2) (2007), 121--126. [Sa] Sahin, B.: Non existence of warped product semi-slant submanifolds of Kaehler manifolds. Geom. Dedicata {\bf 117} (2006), 195--202. [C2] Chen, B.-Y.: \textit{Differential Geometry of Real Submanifolds in a Kaehler Manifold}. Monatsh. Math. {\bf 91} (1981), 257--275. [Se] Sekigawa, K.: Some CR-submanifolds in a $6$-dimensional sphere. Tensor (New Ser.) {\bf 41} (1984), 13--20.

Keywords: nearly Kaehler manifold, warped product, slant submanifold, semi-slant submanifold, generic warped products

Classification (MSC2000): 53C40, 53C42, 53C15

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