Contact Cr-Warped Product Submanifolds of Nearly Quasi-Sasakian Manifold

Year 2019, Volume: 7 Issue: 1, 140 - 145, 15.04.2019

Shamsur Rahman , Mohd Sadiq Khan Aboo Horaira

Abstract

In the present
paper, we construct contact CR-warped product submanifolds of nearly quasi
Sasakian manifold. We have obtained results on the existence of warped product
CR Submanifolds of nearly quasi Sasakian manifold and discuss the
characterization result. We also construct the inequality  for contact CR warped products of nearly quasi
Sasakian manifolds. The equality cases are also discussed.

Keywords

Contact CR-submanifold, Warped product, quasi-Sasakian manifold and Nearly quasi-Sasakian manifold

References

• [1] A. Bejancu, CR-submanifolds of a Kahler manifold. I. Proc. Amer. Math. Soc. 1978, 69 (1), 135–142. doi:10.1090/S0002-9939-1978-0467630-0.
• [2] A. Bejancu and N. Papaghiuc, Semi-invariant submanifolds of a Sasakian manifold. An St Univ Al I Cuza Iasi supl 1981; XVII 1 I-a : 163-170.
• [3] B. H. Kim, Fibred Riemannian spaces with quasi-Sasakian structure, Hiroshima Math. J. 20, 477–513, 1990.
• [4] B. Sahin, Nonexistence of warped products semi-slant submanifolds of Kaehler manifolds, Geometriae Dedicata. 117 (2006) 195-202.
• [5] B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds I, K. Monatsh. Math. 133(2001), 177-195.
• [6] B. Y. Chen, Geometry of warped product CR-Submanifolds in Kaehler Manifolds II, Monatsh. Math. 134 (2001) 103-119.
• [7] B. Y. Chen and M. I. Munteanu, Geometry of PR-warped products in para-Kaehler manifolds, Taiwan. J. Math., 16 (2012), 1293-1327.
• [8] D. E. Blair, The theory of quasi-Sasakian structure, J. Differential Geo. 1, 331-345, 1967.
• [9] I. Hasegawa and I. Mihai, Contact CR-warped product submanifolds in Sasakian manifolds, Geom. Dedicata, 102 (2003), 143-150.
• [10] J. A. Oubina, New classes of almost contact metric structures, Publ. Math. Debrecen 32, 187–193, 1985.
• [11] J. C. Gonzalez, and D. Chinea, Quasi-Sasakian homogeneous structures on the generalized Heisenberg group H(p, 1), Proc. Amer. Math. Soc. 105, 173–184, 1989.
• [12] K. Arslan, R. Ezentas, I. Mihai and C. Murathan, Contact CR-warped product submanifolds in Kenmotsu space forms, J. Korean Math. Soc., 42 (2005), 1101-1110.
• [13] R. L. Bishop and B. O. Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc., 145 (1969), 1-49.
• [14] S. Hiepko, Eine inner kennzeichungder verzerrten produkte, Math. Ann., 241 (1979), 209-215.
• [15] S. Kanemaki, Quasi-Sasakian manifolds, Tohoku Math. J. 29, 227–233, 1977.
• [16] S. Kanemaki, On quasi-Sasakian manifolds, Differential Geometry Banach Center Publications 12, 95–125, 1984.
• [17] S. Rahman and Shafiullah, Geometry of Hypersurfaces of a Semi Symmetric Semi Metric Connection in a Quasi-Sasakian Manifold. Journal of Purvanchal Academy of Sciences, Vol. 17 (2011) pp. 231-242.
• [18] S. Rahman and A. Ahmad, On The Geometry of Hypersurfaces of a Certain Connection in a Quasi-Sasakian Manifold, International Journal Mathematical Combinatorics Vol.3 (2011), pp. 23-33.
• [19] S. Rahman, Some Properties of Hyperbolic contact Manifold in a Quasi Sasakian Manifold, Turkic World Mathematical Society Journal of Applied and Engineering Mathematics Vol. 1 No. 1, (2011), pp. 41-48.
• [20] S. Rahman, Geometry of Hypersurfaces of a semi symmetric metric connection in a quasi-Sasakian manifold, Journal-Proceedings of the Institute of Applied Mathematics, Vol.3 No.2 (2014), pp.152-164.
• [21] S. Rahman, Geometry of hypersurfaces of a quarter semi symmetric non metric connection in a quasi-Sasakian manifold. Carpathian Mathematical Publications Vol. 7(2) (2015) pp. 226-235 doi:10.15330/cmp.7.2.226-235.
• [22] S. Rahman and N. K. Agrawal, On the geometry of slant and pseudo-slant submanifolds in a quasi Sasakian manifolds, J. Modern Technology and Engineering Vol. 2, No.1, 2017, pp.82-89.
• [23] S. Rahman, Contact conformal connection on a geometry of hypersurfaces with certain connection in a quasi-Sasakian manifold, Bulletin of the Transilvania University of Brasov Series III: Mathematics, Informatics, Physics, Vol 10 (59), No. 1, 2017 pp.135-148.
• [24] S. Tanno, Quasi-Sasakian structure of rank 2p + 1, J. Differential Geom. 5, 317–324, 1971.