On 3-dimensional Trans-Sasakian manifold admitting a semi symmetric metric connection

Abstract

The purpose of the present paper is to study  3-dimensional trans-Sasakian manifold admitting a semi-symmetric metric connection. Here we mainly study locally -symmetric and locally -concircularly symmetric 3-dimensional trans-Sasakian manifold admitting a semi-symmetric metric connection. Moreover, we examine our results and the results of [1] and [2] by constructing some examples. 

Keywords

• Concircular curvature tensor,Semi- symmetric metric,Locally -symmetric,Locally concircularly -symmetric

References

1. \bibitem{AAAS} A. Akbar and A. Sarkar,\,\,\emph{Some Curvature Properties of Trans-Sasakian Manifolds,}\,\, Lobachevskii Journal of Mathematics, 35(2), 2014, 56-64.
2. \bibitem{BBAKR} B. Barua and A.K. Ray,\,\ \emph{Some properties of semi-symmetric metric connetion in a Riemannian manifold,} Indian J. Pure Appl. Math., 16(7), (1985), 726-740.
3. \bibitem{icts} D.E. Blair,\,\ \emph{Contact manifolds in Riemannian geometry,% } Lecture Notes in Mathematics, 509 Springer-Verlag, Berlin, 1976.
4. \bibitem{SDMMT} S. Deshmukh and M.M. Tripathi,\,\,\emph{A note on trans-Sasakian manifolds,}\,\, Math. Slov. 63(6), (2013), 1361-1370.
5. \bibitem{UCDeASarkar} U. C. De and A. Sarkar,\,\ \emph{On Three-Dimensional Trans-Sasakian Manifolds,} Extracta mathematicae, 23(3), 2008, 265-277.
6. \bibitem{UCDMMT} U.C. De and M.M. Tripathi,\,\,\emph{Ricci tensor in 3-dimensional trans-Sasakian manifolds,}\,\, Kyungpook Math. J., 43(2), (2003), 247-255.
7. \bibitem{UCDKD} U.C. De and K. De,\,\,\emph{On a class of three-dimensional Trans-Sasakian manifolds,}\,\, Commun. Korean Math. Soc. 27(4), (2012), 795-808.
8. \bibitem{UCDAB} U.C. De and A. Barman,\,\ \emph{On a type of semisymmetric metric connection on a Riemannian manifold,} Publications De L'institut Math% \'{e}matique Nouvelle s\'{e}rie, tome, 98(112), (2015), 211-218.

9. \bibitem{GrayAHervellaLM} A. Gray, L.M. Hervella,\,\,\emph{The sixteen classes of almost Hermitian manifolds and their linear invariants,} \,\, Ann. Mat. Pura Appl., 123(4), 1980, 35-58.
10. \bibitem{HAHayden} H.A. Hayden.\,\,{\ Subspaces of a space with torsion.}% \,\, Proc. London Math. Soc., 34, 1932, 27-50.
11. \bibitem{KKenmotsu} K. Kenmotsu,\,\,\emph{A class of almost contact Riemannian manifolds,}\,\, T\^{o}hoku Math. J., 24, (1972), 93-103.
12. \bibitem{KHDDAB} Kalyan Halder, Dipankar Debnath and Arindam Bhattacharyya,\,\ \emph{Semi-Symmetric Metric Connection on a 3-Dimensional Trans-Sasakian Manifold,} International J.Math. Combin., 3, (2013), 16-21.
13. \bibitem{JSKRPMMT} J.S. Kim, R. Prasad and M.M. Tripathi,\,\,\emph{On generalized Ricci-recurrent trans-Sasakian manifolds,}\,\, J. Korean Math. Soc., 39(6), (2002), 953-961.
14. \bibitem{VFK} V.F. Kirichenko,\,\,\emph{On the geometry of nearly trans-Sasakian manifolds,}\,\, Dokl Akad. Nauk 397(6), (2004), 733-736.
15. \bibitem{MarreroJC} J.C. Marrero,\thinspace \thinspace \emph{The local structure of trans-Sasakian manifolds,}\thinspace \thinspace\ Ann. Mat. Pure Appl., 162, 1992, 77-86.
16. %\bibitem{HGN} H.G. Nagaraja,\,\,{\em $\phi$-recurrent Trans-Sasakian manifolds,}\,\, Matematiqki Vesnik, 63(2), (2011), 79-86.
17. \bibitem{OubinaJA} J.A. Oubina,\,\,\emph{New classes of almost contact metric structures,}\,\, Publ. Math. Debr., 32(3--4), 1985, 187-193.
18. \bibitem{DGPCSBV} D.G. Prakasha, C.S. Bagewadi and Venkatesha,\,\,\emph{% Conformally and quasi-conformally conservative curvature tensors on a trans-Sasakian manifold with respect to semi-symmetric metric connections,}% \,\, Diff. Geometry-Dyn.Sys., 10, (2008), 263-274.
19. \bibitem{CSBDGPV} D.G. Prakasha, C.S. Bagewadi and Venkatesha,\,\,\emph{% Conservative Projective Curvature Tensor On Trans-sasakian Manifolds With Respect To Semi-symmetric Metric Connection,}\,\, An. S% %TCIMACRO{\U{b8}}% %BeginExpansion \c{}% %EndExpansion t. Univ. Ovidius Constanta, 15(2), 2007, 5-18.
20. \bibitem{ASPAB} A. Sampa Pahan and Arindam Bhattacharyya,\,\ \emph{Some Properties of Three Dimensional Trans-Sasakian Manifolds with a Semi-Symmetric Metric Connection,} Lobachevskii Journal of Mathematics, 37(2), 2016, 177-184.
21. \bibitem{AASKKBSE} A.A. Shaikh, K.K. Baishya and S. Eyasmin,\,\,\emph{On $D$% -homothetic deformation of trans-Sasakian structure,}\,\, Demonstratio Mathematica, 41(1), 2008, 171-188.
22. \bibitem{ASSKHMS} A. Sarkar, S.K. Hui and Matilal Sen,\,\,\emph{A Study on Legendre Curves in 3-Dimensional Trans-Sasakian Manifolds,}\,\, Lobachevskii Journal of Mathematics, 35(1), 2014, 11-18.
23. \bibitem{AASSKH} A.A. Shaikh and S.K. Hui,\,\,\emph{On weak symmetries of trans-Sasakian manifolds,}\,\, Proceedings of the Estonian Academy of Sciences, 58(4), 2009, 213-223.
24. \bibitem{tvs10} A. Turgut Vanli and R. Sari,\,\,\emph{ Invariant submanifolds of trans-Sasakian manifolds,}\,\, Differ. Geom. Dyn. Syst. 12, 2010, 277-288.
25. \bibitem{tvs11} A. Turgut Vanli and R. Sari,\,\,\emph{On invariant submanifolds of a nearly trans-Sasakian manifold,}\,\, \,\, Arab. J. Sci. Eng. 36 (3) 2011, 423-429.
26. \bibitem{KYano1} K. Yano,\thinspace \thinspace \emph{Concircular geometry I. Concircular transformations,}\thinspace \thinspace\ Proc. Imp. Acad. Tokyo 16, (1940), 195-200.
27. \bibitem{KYano2} K. Yano,\,\,\emph{On semi-symmetric metric connections.}% \,\, Rev. Roumaine Math. Pures Appl., 15, 1970, 1579-1586.