A Note on Nearly Hyperbolic Cosymplectic Manifolds

Abstract

Interest in hyperbolic cosymplectic manifolds has increased in recent years. In this paper, pseudo slant submanifolds of a nearly hyperbolic cosymplectic manifold have been explored deeply. In particular, the integrability conditions of distributions on such manifolds have been investigated. So, for a pseudo submanifold M of a nearly hyperbolic cosymplectic manifold M ̃ which is totally geodesic, D_θ,D_( _μ ) distributions with〖 D〗_( _μ )⊕⟨ξ⟩ and D_θ⊕D_( _μ ) are proved integrable but D_( _θ )⊕⟨ξ⟩ or not.

Keywords

• Pseudo slant submanifold
• nearly hyperbolic cosymplectic manifold
• integrability condition

References

1. [1] Upadhyay M.D., Dube K.K., “Almost contact hyperbolic (f,g,η,ξ)-structure”. Acta Math. Acad. Sci. Hungary, 28(1-2): 1–4, (1976).
2. [2] Joshi N. K., Dube K. K., “Semi-invariant submanifold of an almost r-contact hyperbolic metric manifold”. Demonstratio Math., 34(1): 135–143, (2001).
3. [3] Dogan S., Karadag, M., “Slant submanifolds of an almost hyperbolic contact metric manifolds”. Journal of Mathematics and System Science, 4: 285–288, (2014).
4. [4] Uddin S., Wong B. R., Mustafa A. A., “Warped product pseudo slant submanifolds of a nearly cosymplectic manifold”. Abstr. Appl. Anal., Art. ID 420890, 1-13, (2012).
5. [5] Ahmad M., Ali K., “Semi-invariant Submanifolds of Nearly Hyperbolic Cosymplectic Manifolds”. Global Journal of Science Frontier Research Mathematics and Decision Sciences, 13(4), 73–81, (2013).
6. [6] Ahmad M., Ali K., “CR-submanifolds of a nearly hyperbolic cosymplectic manifold”. IOSR Journal of Mathematics (IOSR-JM), 6(3): 74–77, (2013).
7. [7] Khan V. A., Khan M. A., “Pseudo Slant Submanifolds of a Sasakian Manifold”. Indian J. Pure Appl. Math., 38(1): 31–42, (2007).
8. [8] De U. C., Sarkar A., “On pseudo slant submanifolds of Trans-Sasakian Manifolds”. Proc. Est. Acad. Sci., 60(1): 1–11, (2011).

9. [9] Blair D.E., “Riemannian Geometry of contact and symplectic manifolds”, Progress in Mathematics, 203. Birkhauser Boston, Inc., Boston , MA, 2002
10. [10] Fujumoto A., Muto H., “On Cosymplectic manifolds”,Tensor N.S., 28:43-52,(1974).
11. [11] Olszak Z., “On almost Cosymplectic manifolds”. Kodai Math. J., 4:239-250,(1981),
12. [12] Koufogiorgos Th., Tsiclias C., “On the existense of a new class of contact metric manifolds”.Canad. Math. Bull., 43:440-447,(2000).
13. [13] Baikoussis C., Koufogiorgos T., “On a type of contact manifolds”. J. Geom. 46:1-9,(1993).
14. [14] Endo H., “On some properties of almost Cosymplectic manifolds”, An Ştiint.Univ. ‘’Al.I.Cuza’’ Iaşi, Math.42: 79-94, (1996).
15. [15].Goldberg S.I., Yano K., “Integrability of almost cosymplectic structures”. Pasific J.Math.31: 373-382, (1969).