# ON ALMOST r-PARACONTACT RIEMANNIAN MANIFOLD WITH A CERTAIN CONNECTION

• Haseeb, Abdul ;
• Jun, Jae-Bok ;
• Rahman, Shamsur
• Published : 2010.04.30
• 61 15

#### Abstract

In a Riemannian manifold, the existence of a new connection is proved. In particular cases, this connection reduces to several symmetric, semi-symmetric and quarter symmetric connections, even some of them are not introduced so far. So, in this paper, we define a quarter symmetric semi-metric connection in an almost r-paracontact Riemannian manifold and consider invariant, non-invariant and anti-invariant hypersurfaces of an almost r-paracontact Riemannian manifold with that connection.

#### Keywords

almost r-paracontact Riemannian manifold;quarter-symmetric semi-metric connection

#### References

1. T. Adati, Hypersurfaces of almost paracontact Riemannian manifolds, TRU Math. 17 (1981), no. 2, 189-198.
2. M. Ahmad, J. B. Jun, and A. Haseeb, Hypersurfaces of almost r-paracontact Riemannian manifold endowed with a quarter symmetric metric connection, Bull. Korean Math. Soc. 46 (2009), no. 3, 477-487. https://doi.org/10.4134/BKMS.2009.46.3.477
3. O. C. Andonie and D. Smaranda, Certaines connexions semi-symetriques, Tensor (N.S.) 31 (1977), no. 1, 8-12.
4. A. Bucki, Hypersurfaces of almost r-paracontact Riemannian manifolds, Tensor (N.S.) 48 (1989), no. 3, 245-251.
5. A. Bucki and A. Miernowski, Invariant hypersurfaces of an almost r-paracontact manifold, Demonstratio Math. 19 (1986), no. 1, 113-121.
6. B. Y. Chen, Geometry of Submanifolds, Marcel Dekker, New York, 1973.
7. L. P. Eisenhart, Continuous Groups of Transformations, Dover Publications, Inc., New York, 1961.
8. A. Friedmann and J. A. Schouten, uber die Geometrie der halbsymmetrischen Ubertragung, Math. Z. 21 (1924), no. 1, 211-223. https://doi.org/10.1007/BF01187468
9. S. Golab, On semi-symmetric and quarter-symmetric linear connections, Tensor (N.S.) 29 (1975), no. 3, 249-254.
10. Y. Liang, On semi-symmetric recurrent-metric connection, Tensor (N.S.) 55 (1994), no. 2, 107-112.
11. I. Mihai and K. Matsumoto, Submanifolds of an almost r-paracontact Riemannian manifold of P-Sasakian type, Tensor (N.S.) 48 (1989), no. 2, 136-142.
12. R. S. Mishra and S. N. Pandey, On quarter symmetric metric F-connections, Tensor (N.S.) 34 (1980), no. 1, 1-7.
13. S. C. Rastogi, On quarter-symmetric metric connection, C. R. Acad. Bulgare Sci. 31 (1978), no. 7, 811-814.
14. B. G. Schmidt, Conditions on a connection to be a metric connection, Comm. Math. Phys. 29 (1973), 55-59. https://doi.org/10.1007/BF01661152
15. J. Sengupta, U. C. De, and T. Q. Binh, On a type of semi-symmetric non-metric connection on a Riemannian manifold, Indian J. Pure Appl. Math. 31 (2000), no. 12, 1659-1670.
16. L. Tamassy and T. Q. Binh, On the nonexistence of certain Riemannian connections with torsion and of constant curvature, Publ. Math. Debrecen 36 (1989), no. 1-4, 283-288
17. K. Yano, On semi-symmetric metric connection, Rev. Roumaine Math. Pures Appl. 15 (1970), 1579-1586.
18. K. Yano and T. Imai, Quarter-symmetric metric connections and their curvature tensors, Tensor (N.S.) 38 (1982), 13-18.

#### Cited by

1. $$\textit{CR}$$ CR -submanifolds and $$\textit{CR}$$ CR -products of a Lorentzian para-Sasakian manifold endowed with a quarter symmetric semi-metric connection vol.25, pp.4, 2014, https://doi.org/10.1007/s13370-013-0180-4