Abdulqader MUSTAFA, Cenap OZEL, Alexander PİGAZZİNİ, Richard PİNCAK

Existence and Nonexistence of Warped Product Submanifolds of Almost Contact Manifolds

This paper has two goals; the first is to generalize results for the existence and nonexistence of warped product submanifolds of almost contact manifolds, accordingly a self-contained reference of such submanifolds is offered to save efforts of potential research. Most of the results of this paper are general and decisive enough to generalize both known and new results. Moreover, a discrete example of contact $CR$-warped product submanifold in Kenmotsu manifold is constructed. For further research direction, we addressed a couple of open problems arose from the results of this paper.

Keywords:

Contact $CR$-warped products, Sasakian manifolds, Kenmotsu manifolds, cosymplectic manifolds, nearly trans-Sasakian manifolds general warped product, doubly warped product, second fundamental form, totally geodesic,

PDF

___

[1] O’Neill, B.: Semi-Riemannian geometry with applictions to relativity, Academic Press, New York, 1983.
[2] Bishop, R.L., O’Neill, B.: Manifolds of negative curvature, Transactions of the American Mathematical Society, 145, 1-49 (1969).
[3] Carmo, D.M.: Riemannian Geometry, Birkha¨user Boston, (1992).
[4] Bejancu, A.: Geometry of CR-submanifolds, D. Reidel Publishing Company, (1986).
[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 slant submanifolds, Katholieke Universiteit Leuven, Leuven, (1990).
[7] Chen, B.Y.: On warped product immersions, Journal of Geometry, 82, 36-49 (2005).
[8] Chen, B.Y.: Geometry of warped products as Riemannian submanifolds and related problems, Soochow J. Math., 28, 125-156 (2002).
[9] Blair, D.E.: Almost contact manifolds with Killing structure tensors I. Pacific Journal of Mathematics, 39, 285-292 (1971).
[10] Oubina, J.A.: New classes of almost contact metric structures. Publicationes Mathematicae Debrecen, 32, 187-193 (1985).
[11] Sasaki, S.: On differentiable manifolds with certain structures which are closely related to almost contact structure. Tohoku Mathematical Journal, 12, 459-76 (1960).
[12] Kenmotsu, K.: A class of almost contact Riemannian manifolds. Tohoku Mathematical Journal, 24, 93-103 (1972).
[13] Olszak, Z.: On almost cosymplectic manifolds. Kodai Mathematical Journal, 1, 239-250 (1981).
[14] Gray, A., Hervella, L.: The sixteen classes of almost Hermitian manifolds and their linear invariants. Annali di Matematica Pura ed Applicata , 123, 35-58 (1980).
[15] Gherghe, C.: Harmonicity of nearly trans-Sasaki manifolds. Demonstratio Mathematica, 33, 151-157 (2000).
[16] Chinea, D., Gonzalez, C.: A classification of almost contact metric manifolds, Annali di Matematica Pura ed Applicata, 156, 15-36 (1990).
[17] Marrero, J.C.: The local structure of trans-Sasakian manifolds. Annali di Matematica Pura ed Applicata, 162, 77-86 (1992).
[18] Khan, V.A., Khan, K.A., Uddin, S.: Contact CR-warped product submanifolds of Kenmotsu manifolds. Thai Journal of Mathematics, 6(1), 138-145 (2008).
[19] Munteanu, M.I.: Warped product contact CR-submanifolds of Sasakian space forms. Publicationes Mathematicae Debrecen, 66, 75-120 (2005).
[20] Chen, B.Y.: A survey on geometry of warped product submanifolds, J. Adv. Math. Stud. 6, 1-43 (2013).
[21] Mustafa, A., Uddin, S., Khan, V.A., Wong, B.R.: Contact CR-warped product submanifolds of nearly trans-Sasakian manifolds. Taiwanese Journal of Mathematics, 17(4), 1473-1486 (2013).
[22] Uddin, S., Mustafa, A., Wong, B.R., Ozel, C.: A geometric inequality for warped product semi-slant submanifolds of nearly cosymplectic manifolds. Revista de la Unio´n Matema´tica Argentina, 55, 55-69 (2014).
[23] Mustafa, A., Uddin, S., Wong, B.R.: Generalized inequalities on warped product submanifolds in nearly trans-Sasakian manifolds. Journal of Inequalities and Applications, 346 (2014).
[24] Hasegawa, I., Mihai, I.: Contact CR-warped product submanifolds in Sasakian manifolds. Geometriae Dedicata, 102, 143-150 (2003).
[25] Hiepko, S.: Eine inner kennzeichungder verzerrten produkte. Mathematische Annalen, 241, 209-215 (1979).