## Tbilisi Mathematical Journal

### Indefinite trans-Sasakian manifold with semi-symmetric metric connection

#### Abstract

The objective of the present paper is to study of indefinite trans-Sasakian manifold with a semi-symmetric metric connection. We have found the relations between curvature tensors, Ricci curvature tensors and scalar curvature of indefinite trans-Sasakian manifolds with semi-symmetric metric connection and with metric connection. Also, we have proved some results on quasi-projectively flat and $\varphi -$projectively flat manifolds with respect to semi-symmetric metric connection. It is shown that the manifold satisfying $\overset{\_}{R.}\overset{\_}{S}$ $=0$ is an $\eta -$Einstein manifold if $\alpha =0$ and $\beta =constant.$ It is also proved that the manifold satisfying $\overset{\_}{P}.\overset{\_}{S}=0$ is an $\eta -$ Einstein manifold if $\alpha =0$ and $\beta =constant.$ Finally, we have obtained the conditions for the manifold with semi-symmetric metric connection to be conformally flat and $\xi -$conformally flat.

#### Article information

Source
Tbilisi Math. J., Volume 8, Issue 2 (2015), 233-255.

Dates
Accepted: 15 September 2015
First available in Project Euclid: 12 June 2018

https://projecteuclid.org/euclid.tbilisi/1528769020

Digital Object Identifier
doi:10.1515/tmj-2015-0025

Mathematical Reviews number (MathSciNet)
MR3420407

Zentralblatt MATH identifier
1328.53055

#### Citation

Prasad, Rajendra; Kumar, Sushil. Indefinite trans-Sasakian manifold with semi-symmetric metric connection. Tbilisi Math. J. 8 (2015), no. 2, 233--255. doi:10.1515/tmj-2015-0025. https://projecteuclid.org/euclid.tbilisi/1528769020

#### References

• C. S. Bagewadi, On totally real submanifolds of a Kahlerian manifold admitting Semi symmetric metric F-connection. Indian. J. Pure. Appl. Math, 13, 528-536, (1982).
• C. S. Bagewadi and N. B. Gatti, On irrotational quasi-conformal curvature tensor. Tensor.N.S., 64, 284-258, (2003).
• C. S. Bagewadi and E. Girish Kumar, Note on Trans-Sasakian Manifolds. Tensor. N. S., 65, 80-88 (2004).
• A. Bejancu and K. L. Duggal, Real hypersurfaces of idefinite Kahler manifolds. Int. J. Math. Math. sci., 16, 545-556, (1993)..
• D. E. Blair, Contact manifolds in Riemannian geometry. Lecture note in Mathematics, 509, Springer-Verlag Berlin-New York, 1976.
• U. C. De. and A. Sarkar, On $(\epsilon )-$ Kenmotsu manifolds. Hadronic journal, 32, 231-242, (2009).
• U. C. De and Absos Ali Shaikh, K-contact and Sasakian manifolds with conservative quasi-conformal curvature tensor. Bull. Cal. Math. Soc., 89, 349-354, (1997).
• A. Friedmann and J. A. Schouten, Uber die geometric der holbsymmetrischen Ubertragurgen. Math. Zeitschr. 21, 211-233, (1924).
• H. A. Hayden, Subspaces of space with torsion. Proc. Lond. Math. Soc. 34 , 27-50, (1932).
• S. I. Hussain and A. Sharafuddin, Semi-symmetric metric connections in almost contact mani-folds. Tensor, N. S.,30, 133-139, (1976).
• Amur Kumar and S. S. Pujar, On Submanifolds of a Riemannian manifold admitting a metric semi-symmetric connection. Tensor, N. S., 32, 35-38, (1978).
• R. Kumar, R. Rani and R. K. Nagaich, On sectional curvature of $(\epsilon )-$Sasakian manifolds. Int. J. Math. Math. sci., (2007), ID. 93562.
• J. C. Marrero, The local structures of trans-Sasakian manifolds. Ann. Mat. Pura. Appl, (4), 162, 77-86, (1992).
• Halammanavar G. Nagaraja, Rangaswami C. Premalatha and Ganganna Somashekara, On an $(\epsilon ,\delta )$ trans-Sasakian structure. Proceedins of the Estonian Academy of Sciences, (1), 61, 20-28 (2012).
• J. A. $Oubi\overset{\thicksim }{na}$, New classes of almost contact metric structures. Publicationes Mathematicae Debrecen,vol. 32, 187-193, (1985).
• M. M. Tripathi, Ricci solitons in contact metric manifolds . arXiv:0801.4222v1,[math.DG],28, (2008).
• K. Yano, On semi-symmetric metric connections. Revue Roumaine de Math. Pures et Appliques., 15 1579-1586, (1970).
• X.Xufeng and C. Xiaoli, Two theorems on $(\epsilon )-$ Sasakian manifolds. Int. J. Math. Sci., 21 249 -254, (1998).