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Indefinite trans-Sasakian manifold with semi-symmetric metric connection. (English) Zbl 1328.53055
Summary: The objective of the present paper is to study indefinite trans-Sasakian manifold with a semi-symmetric metric connection. We find 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 prove some results on quasi-projective and \(\varphi\)-projective manifolds with respect to semi-symmetric metric connection. It is shown that the manifold satisfying \(\bar{R}:\bar{S} = 0\) is an \(\eta\)-Einstein manifold if \(\alpha = 0\) and \(\beta =\) constant. It is also proved that the manifold satisfying \(\bar{P}:\bar{S} = 0\) is an \(\eta\)-Einstein manifold if \(\alpha = 0\) and \(\beta =\) constant. Finally, we obtain the conditions for the manifold with semi-symmetric metric connection to be conformally at and \(\xi\)-conformally flat.
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
Full Text: DOI
[1] [1] C. S. Bagewadi, On totally real submanifolds of a Kahlerian manifold admitting Semi symmet- ric metric F-connection. Indian. J. Pure. Appl. Math, 13, 528-536, (1982) . · Zbl 0482.53050
[2] C. S. Bagewadi and N. B. Gatti, On irrotational quasi-conformal curvature tensor. Tensor.N.S., 64, 284-258, (2003). · Zbl 1165.53371
[3] C. S. Bagewadi and E. Girish Kumar, Note on Trans-Sasakian Manifolds. Tensor. N. S., 65, 80-88 (2004). · Zbl 1165.53343
[4] A. Bejancu and K. L. Duggal, Real hypersurfaces of idefinite Kahler manifolds. Int. J. Math. Math. sci., 16, 545-556, (1993). . · Zbl 0787.53048
[5] D. E. Blair, Contact manifolds in Riemannian geometry. Lecture note in Mathematics, 509, Springer-Verlag Berlin-New York, 1976. · Zbl 0319.53026
[6] U. C. De. and A. Sarkar, On (”) Kenmotsu manifolds. Hadronic journal, 32 , 231-242, (2009). · Zbl 1195.53065
[7] 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). · Zbl 0905.53035
[8] A. Friedmann and J. A. Schouten, Uber die geometric der holbsymmetrischen Ubertragurgen. Math. Zeitschr. 21, 211-233, (1924).
[9] H. A. Hayden, Subspaces of space with torsion. Proc. Lond. Math. Soc. 34 , 27-50, (1932) . · Zbl 0005.26601
[10] S. I. Hussain and A. Sharafuddin, Semi-symmetric metric connections in almost contact mani- folds. Tensor, N. S.,30, 133-139, (1976).
[11] 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). · Zbl 0379.53004
[12] R. Kumar, R. Rani and R. K. Nagaich, On sectional curvature of (”)Sasakian manifolds. Int. J. Math. Math. sci., (2007), ID. 93562. · Zbl 1141.53307
[13] J. C. Marrero, The local structures of trans-Sasakian manifolds. Ann. Mat. Pura. Appl, (4), 162, 77-86, (1992). · Zbl 0772.53036
[14] Halammanavar G. Nagaraja, Rangaswami C. Premalatha and Ganganna Somashekara, On an (ε;δ) trans-Sasakian structure. Proceedins of the Estonian Academy of Sciences, (1), 61, 20-28 (2012). · Zbl 1244.53035
[15] J. A. Oubi s na, New classes of almost contact metric structures. Publicationes Mathematicae Debrecen ,vol.32, 187-193, (1985).
[16] M. M. Tripathi, Ricci solitons in contact metric manifolds. arXiv:0801.4222v1, [math.DG],28, (2008).
[17] K. Yano, On semi-symmetric metric connections. Revue Roumaine de Math. Pures et Ap- pliques., 15 1579-1586, (1970) . · Zbl 0213.48401
[18] X.Xufeng and C. Xiaoli, Two theorems on (”)Sasakian manifolds. Int. J. Math. Sci., 21 249 -254, (1998). · Zbl 0901.53050
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