Year 2019,
Volume: 32 Issue: 1, 242 - 254, 01.03.2019
Srivaishnava Vasudeva Vıshnuvardhana
,
Venkatesh Venkatesha
,
Aysel Turgut Vanlı
References
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Trans-Sasakian Manifolds,}\,\, Lobachevskii Journal of Mathematics, 35(2),
2014, 56-64.
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semi-symmetric metric connetion in a Riemannian manifold,} Indian J. Pure
Appl. Math., 16(7), (1985), 726-740.
- \bibitem{icts} D.E. Blair,\,\ \emph{Contact manifolds in Riemannian geometry,%
} Lecture Notes in Mathematics, 509 Springer-Verlag, Berlin, 1976.
- \bibitem{SDMMT} S. Deshmukh and M.M. Tripathi,\,\,\emph{A note on
trans-Sasakian manifolds,}\,\, Math. Slov. 63(6), (2013), 1361-1370.
- \bibitem{UCDeASarkar} U. C. De and A. Sarkar,\,\ \emph{On Three-Dimensional
Trans-Sasakian Manifolds,} Extracta mathematicae, 23(3), 2008, 265-277.
- \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.
- \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.
- \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.
- \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.
- \bibitem{HAHayden} H.A. Hayden.\,\,{\ Subspaces of a space with torsion.}%
\,\, Proc. London Math. Soc., 34, 1932, 27-50.
- \bibitem{KKenmotsu} K. Kenmotsu,\,\,\emph{A class of almost contact
Riemannian manifolds,}\,\, T\^{o}hoku Math. J., 24, (1972), 93-103.
- \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.
- \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.
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trans-Sasakian manifolds,}\,\, Dokl Akad. Nauk 397(6), (2004), 733-736.
- \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.
- %\bibitem{HGN} H.G. Nagaraja,\,\,{\em $\phi$-recurrent Trans-Sasakian manifolds,}\,\, Matematiqki Vesnik, 63(2), (2011), 79-86.
- \bibitem{OubinaJA} J.A. Oubina,\,\,\emph{New classes of almost contact
metric structures,}\,\, Publ. Math. Debr., 32(3--4), 1985, 187-193.
- \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.
- \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.
- \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.
- \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.
- \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.
- \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.
- \bibitem{tvs10} A. Turgut Vanli and R. Sari,\,\,\emph{ Invariant submanifolds of trans-Sasakian manifolds,}\,\, Differ. Geom. Dyn. Syst. 12, 2010, 277-288.
- \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.
- \bibitem{KYano1} K. Yano,\thinspace \thinspace \emph{Concircular geometry I.
Concircular transformations,}\thinspace \thinspace\ Proc. Imp. Acad. Tokyo
16, (1940), 195-200.
- \bibitem{KYano2} K. Yano,\,\,\emph{On semi-symmetric metric connections.}%
\,\, Rev. Roumaine Math. Pures Appl., 15, 1970, 1579-1586.
On 3-dimensional Trans-Sasakian manifold admitting a semi symmetric metric connection
Year 2019,
Volume: 32 Issue: 1, 242 - 254, 01.03.2019
Srivaishnava Vasudeva Vıshnuvardhana
,
Venkatesh Venkatesha
,
Aysel Turgut Vanlı
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.
References
- \bibitem{AAAS} A. Akbar and A. Sarkar,\,\,\emph{Some Curvature Properties of
Trans-Sasakian Manifolds,}\,\, Lobachevskii Journal of Mathematics, 35(2),
2014, 56-64.
- \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.
- \bibitem{icts} D.E. Blair,\,\ \emph{Contact manifolds in Riemannian geometry,%
} Lecture Notes in Mathematics, 509 Springer-Verlag, Berlin, 1976.
- \bibitem{SDMMT} S. Deshmukh and M.M. Tripathi,\,\,\emph{A note on
trans-Sasakian manifolds,}\,\, Math. Slov. 63(6), (2013), 1361-1370.
- \bibitem{UCDeASarkar} U. C. De and A. Sarkar,\,\ \emph{On Three-Dimensional
Trans-Sasakian Manifolds,} Extracta mathematicae, 23(3), 2008, 265-277.
- \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.
- \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.
- \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.
- \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.
- \bibitem{HAHayden} H.A. Hayden.\,\,{\ Subspaces of a space with torsion.}%
\,\, Proc. London Math. Soc., 34, 1932, 27-50.
- \bibitem{KKenmotsu} K. Kenmotsu,\,\,\emph{A class of almost contact
Riemannian manifolds,}\,\, T\^{o}hoku Math. J., 24, (1972), 93-103.
- \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.
- \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.
- \bibitem{VFK} V.F. Kirichenko,\,\,\emph{On the geometry of nearly
trans-Sasakian manifolds,}\,\, Dokl Akad. Nauk 397(6), (2004), 733-736.
- \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.
- %\bibitem{HGN} H.G. Nagaraja,\,\,{\em $\phi$-recurrent Trans-Sasakian manifolds,}\,\, Matematiqki Vesnik, 63(2), (2011), 79-86.
- \bibitem{OubinaJA} J.A. Oubina,\,\,\emph{New classes of almost contact
metric structures,}\,\, Publ. Math. Debr., 32(3--4), 1985, 187-193.
- \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.
- \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.
- \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.
- \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.
- \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.
- \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.
- \bibitem{tvs10} A. Turgut Vanli and R. Sari,\,\,\emph{ Invariant submanifolds of trans-Sasakian manifolds,}\,\, Differ. Geom. Dyn. Syst. 12, 2010, 277-288.
- \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.
- \bibitem{KYano1} K. Yano,\thinspace \thinspace \emph{Concircular geometry I.
Concircular transformations,}\thinspace \thinspace\ Proc. Imp. Acad. Tokyo
16, (1940), 195-200.
- \bibitem{KYano2} K. Yano,\,\,\emph{On semi-symmetric metric connections.}%
\,\, Rev. Roumaine Math. Pures Appl., 15, 1970, 1579-1586.