Research Article
BibTex RIS Cite
Year 2023, , 367 - 378, 30.04.2023
https://doi.org/10.36890/iejg.1131334

Abstract

References

  • [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).

Existence and Nonexistence of Warped Product Submanifolds of Almost Contact Manifolds

Year 2023, , 367 - 378, 30.04.2023
https://doi.org/10.36890/iejg.1131334

Abstract

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.

References

  • [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).
There are 25 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Abdulqader Mustafa 0000-0001-8380-4562

Cenap Ozel 0000-0001-8005-7039

Alexander Pigazzini 0000-0002-8509-7512

Richard Pincak 0000-0002-5772-4777

Early Pub Date April 26, 2023
Publication Date April 30, 2023
Acceptance Date December 10, 2022
Published in Issue Year 2023

Cite

APA Mustafa, A., Ozel, C., Pigazzini, A., Pincak, R. (2023). Existence and Nonexistence of Warped Product Submanifolds of Almost Contact Manifolds. International Electronic Journal of Geometry, 16(1), 367-378. https://doi.org/10.36890/iejg.1131334
AMA Mustafa A, Ozel C, Pigazzini A, Pincak R. Existence and Nonexistence of Warped Product Submanifolds of Almost Contact Manifolds. Int. Electron. J. Geom. April 2023;16(1):367-378. doi:10.36890/iejg.1131334
Chicago Mustafa, Abdulqader, Cenap Ozel, Alexander Pigazzini, and Richard Pincak. “Existence and Nonexistence of Warped Product Submanifolds of Almost Contact Manifolds”. International Electronic Journal of Geometry 16, no. 1 (April 2023): 367-78. https://doi.org/10.36890/iejg.1131334.
EndNote Mustafa A, Ozel C, Pigazzini A, Pincak R (April 1, 2023) Existence and Nonexistence of Warped Product Submanifolds of Almost Contact Manifolds. International Electronic Journal of Geometry 16 1 367–378.
IEEE A. Mustafa, C. Ozel, A. Pigazzini, and R. Pincak, “Existence and Nonexistence of Warped Product Submanifolds of Almost Contact Manifolds”, Int. Electron. J. Geom., vol. 16, no. 1, pp. 367–378, 2023, doi: 10.36890/iejg.1131334.
ISNAD Mustafa, Abdulqader et al. “Existence and Nonexistence of Warped Product Submanifolds of Almost Contact Manifolds”. International Electronic Journal of Geometry 16/1 (April 2023), 367-378. https://doi.org/10.36890/iejg.1131334.
JAMA Mustafa A, Ozel C, Pigazzini A, Pincak R. Existence and Nonexistence of Warped Product Submanifolds of Almost Contact Manifolds. Int. Electron. J. Geom. 2023;16:367–378.
MLA Mustafa, Abdulqader et al. “Existence and Nonexistence of Warped Product Submanifolds of Almost Contact Manifolds”. International Electronic Journal of Geometry, vol. 16, no. 1, 2023, pp. 367-78, doi:10.36890/iejg.1131334.
Vancouver Mustafa A, Ozel C, Pigazzini A, Pincak R. Existence and Nonexistence of Warped Product Submanifolds of Almost Contact Manifolds. Int. Electron. J. Geom. 2023;16(1):367-78.