Existence of Warped Product Submanifolds of Almost Hermitian Manifolds

Year 2021, Volume: 14 Issue: 2, 361 - 370, 29.10.2021

Abdulqader Mustafa , Cenap Özel

Abstract

This paper has two goals; the first is to generalize results for the existence and nonexistence of warped product submanifolds of almost Hermitian manifolds, accordingly a self-contained reference of such submanifolds is offered to save efforts of other researchers, which is the second goal. At the end of the paper a list of warped products is tabulated whether exist or not. Moreover, a discrete example of $CR$-warped product submanifold in Kaehler manifold is constructed. For further research direction, we addressed a couple of open problems arose from the results of this paper.

Keywords

CR-warped products, Kaehler, nearly Kaehler, general warped product, doubly warped product, second fundamental form, totally geodesic

Thanks

The first author (Abdulqader Mustafa) would like to thank the Palestine Technical University Kadoori, PTUK, for its supports to accomplish this work.

References

• [1] Bejancu, A.: CR submanifolds of a Kaehler manifold I, Proc. Amer. Math. Soc. 69, 135-142 (1978).
• [2] Bejancu, A. : Geometry of CR-submanifolds. D. Reidel Publishing Company (1986).
• [3] Bishop, R. L., O’Neill, B. : Manifolds of negative curvature, Transactions of the American Mathematical Society. 145, 1-49 (1969).
• [4] Chen, B. Y. : Geometry of slant submanifolds. Katholieke Universiteit Leuven, Leuven (1990).
• [5] Chen, B. Y.: Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasgow Math. J. 41), 33-41 (1999).
• [6] Chen, B. Y.: Geometry of warped product CR-submanifolds in Kaehler manifolds. Monatshefte für Mathematik, 133, 177-195 (2001).
• [7] Chen, B. Y.:Geometry of warped products as Riemannian submanifolds and related problems, Soochow J. Math. 28, 125-156 (2002).
• [8] Chen, B. Y.: On warped product immersions, Journal of Geometry, 82 (1-2), 36-40 (2005).
• [9] Chen, B. Y.: A survey on geometry of warped product submanifolds, J. Adv. Math. Stud. 6, (2), 1-43 (2013).
• [10] Do Carmo, M. : Riemannian Geometry. Birkhauser. Boston (1992).
• [11] Khan, V. A. & Khan, K. A. : Generic warped product submanifolds in nearly Kaehler manifolds. Contributions to Algebra and Geometry. 50(2), 337-352 (2009).
• [12] Moroianu, A. : Lectures on Kaehler Geometry, Cambridge University Press (2007).
• [13] Mustafa, A., De, A. and Uddin, S. : Characterization of warped product submanifolds in Kenmotsu manifolds, Balkan J. Geom. Appl. 20 (1), 86-97 (2015).
• [14] Nash, J. F.: C1-isometric imbeddings , Annals Math. 60(3), 383-396 (1954).
• [15] Nash, J. F.: The imbedding problem for Riemannian manifolds, Annals Math. 63(1), 20-63 (1956).
• [16] O’Neill, B. : Semi-Riemannian geometry with applications to relativity. Academic Press. New York (1983).
• [17] Sahin, B. : Non-existence of warped product semi-slant submanifolds of Kaehler manifolds. Geometriae Dedicata,117,195-202 (2006).