ON A CLASS OF THREE-DIMENSIONAL TRANS-SASAKIAN MANIFOLDS

Title & Authors
ON A CLASS OF THREE-DIMENSIONAL TRANS-SASAKIAN MANIFOLDS
De, Uday Chand; De, Krishnendu;

Abstract
The object of the present paper is to study 3-dimensional trans-Sasakian manifolds with conservative curvature tensor and also 3-dimensional conformally flat trans-Sasakian manifolds. Next we consider compact connected $\small{\eta}$-Einstein 3-dimensional trans-Sasakian manifolds. Finally, an example of a 3-dimensional trans-Sasakian manifold is given, which verifies our results.
Keywords
trans-Sasakian manifold;conservative curvature tensor;$\small{\eta}$-Einstein manifold;
Language
English
Cited by
References
1.
C. S. Bagewadi and Venkatesha, Some curvature tensors on a trans-sasakian manifolds, Turkish J. Math. 31 (2007), no. 2, 111-121.

2.
D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Note in Mathematics, Vol. 509, Springer-Verlag, Berlin-New York, 1976.

3.
D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Mathematics, Vol. 203, Birkhauser Boston, Inc., Boston, 2002.

4.
D. E. Blair and J. A. Oubina, Conformal and related changes of metric on the product of two almost contact metric manifolds, Publ. Mat. 34 (1990), no. 1, 199-207.

5.
C. P. Boyer and K. Galicki, Einstein manifolds and contact geometry, Proc. Amer. Math. Soc. 129 (2001), no. 8, 2419-2430.

6.
D. Chinea and C. Gonzales, A classification of almost contact metric manifolds, Ann. Mat. Pura Appl. (4) 156 (1990), 15-36.

7.
D. Chinea and C. Gonzales, Curvature relations in trans-sasakian manifolds, in "Proceedings of the XIIth Portuguese-Spanish Conference on Mathematics, Vol.II,(Portuguese), Braga, 1987", Univ. Minho, Braga, (1987), 564-571.

8.
U. C. De and A. Sarkar, On three-dimensional trans-Sasakian manifolds, Extracta Math. 23 (2008), no. 3, 265-277.

9.
U. C. De and M. M. Tripathi, Ricci tensor in 3-dimensional trans-Sasakian manifolds, Kyungpook Math. J. 43 (2003), no. 2, 247-255.

10.
A. Gray and L. M. Hervella, The sixteen classes of almost hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. (4) 123 (1980), 35-58.

11.
N. J. Hicks, Notes on Differential Geometry, Affilated East-West Press Pvt. Ltd. 1965.

12.
D. Janssens and L. Vanhecke, Almost contact structures and curvature tensors, Kodai Math. J. 4 (1981), no. 1, 1-27.

13.
J. B. Jun and U. K. Kim, On 3-dimensional almost contact metric manifolds, Kyungpook Math. J. 34 (1994), no. 2, 293-301.

14.
J. S. Kim, R. Prasad, and M. M. Tripathi, On generalized Ricci-recurrent trans-Sasakian manifolds, J. Korean Math. Soc. 39 (2002), no. 6, 953-961.

15.
J. C. Marrero, The local structure of trans-sasakian manifolds, Ann. Mat. Pura Appl. (4) 162 (1992), 77-86.

16.
J. C. Marrero and D. Chinea, On trans-Sasakian manifolds, in "Proceedings of the XIVth Spanish-Portuguese Conference on Mathematics, Vol.I-III, (Spanish),Puerto de la Cruz, 1989", Univ. La Laguna, La Laguna (1990), 655-659.

17.
J. A. Oubina, New classes of almost contact metric structures, Publ. Math. Debrecen 32 (1985), no. 3-4, 187-193.

18.
S. S. Shukla and D. D. Singh, On ${\epsilon}$-trans-Sasakian manifolds, Int. J. Math. Anal. (Ruse) 4 (2010), no. 49-52, 2401-2414.

19.
K. Yano, Integral Formulas in Riemannian Geometry, Mercel Dekker, INC., New York, 1970.