On locally $\phi$-semisymmetric Sasakian manifolds

Absos Ali Shaikh; Chandan Kumar Mondal; Helaluddin Ahmad

Research Article

Year 2019, Volume: 48 Issue: 4, 1156 - 1169, 08.08.2019

Absos Ali Shaikh Chandan Kumar Mondal Helaluddin Ahmad

Abstract

References

[1] D.E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math., Springer-Verlag, 1976.
[2] É. Cartan, Sur une classe remarquable déspaces de Riemann, I, Bull. de la Soc. Math. de France 54, 214-216, 1926.
[3] É. Cartan, Sur une classe remarquable déspaces de Riemann, II, Bull. de la Soc. Math. de France 55, 114-134, 1927.
[4] É. Cartan, Lecons sur la geometrie des espaces de Riemann, 2nd ed., Paris, 1946.
[5] M.C. Chaki and M. Tarafdar, On a type of Sasakian manifold, Soochow J. Math. 16, 23-28, 1990.
[6] U.C. De, A.A. Shaikh and S. Biswas, On $\phi$-recurrent Sasakian manifolds, Novi Sad J. Math. 33(2), 43-48, 2003.
[7] K. Ogiue, On fiberings of almost contact manifolds, Kodai Math. Sem. Rep. 17, 53-62, 1965.
[8] A.A. Shaikh and K.K. Baishya, On $\phi$-symmetric LP-Sasakian manifolds, Yokohama Math. J. 52, 97-112, 2005.
[9] A.A. Shaikh, K.K. Baishya and S. Eyasmin, On $\phi$-recurrent generalized $(k, \mu)$-contact metric manifolds, Lobachevski J. Math. 27, 3-13, 2007.
[10] A.A. Shaikh, K.K. Baishya and S. Eyasmin, On the existence of some types of LP-Sasakian manifolds, Commun. Korean Math. Soc., 23 (1), 1-16, 2008.
[11] A.A. Shaikh, T. Basu and S. Eyasmin, On locally $\phi$-symmetric $(LCS)_n$-manifolds, Int. J. Pure Appl. Math. 41 (8), 1161-1170, 2007.
[12] A.A. Shaikh, T. Basu and S. Eyasmin, On the existence of ϕ-recurrent $(LCS)_n$- manifolds, Extracta Mathematica 23 (1), 71-83, 2008.
[13] A.A. Shaikh and U.C. De, On 3-dimensional LP-Sasakian manifolds, Soochow J. Math. 26 (4), 359-368, 2000.
[14] A.A. Shaikh and S.K. Hui, On locally $\phi$-symmetric $\beta$-Kenmotsu manifolds, Extracta Mathematica 24 (3), 301-316, 2010.
[15] A.A. Shaikh and S.K. Hui, On extended $\phi$-recurrent $\beta$-Kenmotsu manifolds, Publi. de l’ Inst. Math., Nouvelle serie, 89 (103), 77-88, 2011.
[16] A.A. Shaikh and H. Kundu, On equivalency of various geometric structures, J. Geom. 105, 139-165, 2014.
[17] Z.I. Szabó, Structure theorems on Riemannian spaces satisfying $R(X,Y)\cdot R=0$, I, The local version, J. Diff. Geom. 17, 531-582, 1982.
[18] Z.I. Szabó, Structure theorems on Riemannian spaces satisfying $R(X,Y)\cdot R=0$, II, Global version, Geom. Dedicata 19, 65-108, 1983.
[19] Z.I. Szabó, Classification and construction of complete hypersurfaces satisfying $R(X,Y)\cdot R=0$, Acta. Sci. Math. 47, 321-348, 1984.
[20] T. Takahashi, Sasakian $\phi$-symmetric spaces, Tohoku Math. J. 29, 91-113, 1977.
[21] S. Tanno, Isometric immersions of Sasakian manifold in spheres, Kodai Math. Sem. Rep. 21, 448-458, 1969.
[22] K. Yano and M. Kon, Structures on manifolds, World Scientific Publ., Singapore, 1984.
[23] H. Weyl, Reine infinitesimal geometrie, Math. Zeitschrift 2, 384-411, 1918.

On locally $\phi$-semisymmetric Sasakian manifolds

Year 2019, Volume: 48 Issue: 4, 1156 - 1169, 08.08.2019

Absos Ali Shaikh Chandan Kumar Mondal Helaluddin Ahmad

Abstract

Generalizing the notion of local $\phi$-symmetry of Takahashi [Sasakian $\phi$-symmetric spaces, Tohoku Math. J., 1977], in the present paper, we introduce the notion of \textit{local $\phi$-semisymmetry} of a Sasakian manifold along with its proper existence and characterization. We also study the notion of local Ricci (resp., projective, conformal) $\phi$-semisymmetry of a Sasakian manifold and obtain its characterization. It is shown that the local $\phi$-semisymmetry, local projective $\phi$-semisymmetry and local concircular $\phi$-semisymmetry are equivalent. It is also shown that local conformal $\phi$-semisymmetry and local conharmonical $\phi$-semisymmetry are equivalent.

Keywords

Sasakian manifold, locally $\phi$-symmetric, $\phi$-semisymmetric, Ricci $\phi$-semisymmetric, projectively $\phi$-semisymmetric, conformally $\phi$-semisymmetric, manifold of constant curvature, $B$-tensor, $B$-$\phi$-semisymmetric

References

[1] D.E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math., Springer-Verlag, 1976.
[2] É. Cartan, Sur une classe remarquable déspaces de Riemann, I, Bull. de la Soc. Math. de France 54, 214-216, 1926.
[3] É. Cartan, Sur une classe remarquable déspaces de Riemann, II, Bull. de la Soc. Math. de France 55, 114-134, 1927.
[4] É. Cartan, Lecons sur la geometrie des espaces de Riemann, 2nd ed., Paris, 1946.
[5] M.C. Chaki and M. Tarafdar, On a type of Sasakian manifold, Soochow J. Math. 16, 23-28, 1990.
[6] U.C. De, A.A. Shaikh and S. Biswas, On $\phi$-recurrent Sasakian manifolds, Novi Sad J. Math. 33(2), 43-48, 2003.
[7] K. Ogiue, On fiberings of almost contact manifolds, Kodai Math. Sem. Rep. 17, 53-62, 1965.
[8] A.A. Shaikh and K.K. Baishya, On $\phi$-symmetric LP-Sasakian manifolds, Yokohama Math. J. 52, 97-112, 2005.
[9] A.A. Shaikh, K.K. Baishya and S. Eyasmin, On $\phi$-recurrent generalized $(k, \mu)$-contact metric manifolds, Lobachevski J. Math. 27, 3-13, 2007.
[10] A.A. Shaikh, K.K. Baishya and S. Eyasmin, On the existence of some types of LP-Sasakian manifolds, Commun. Korean Math. Soc., 23 (1), 1-16, 2008.
[11] A.A. Shaikh, T. Basu and S. Eyasmin, On locally $\phi$-symmetric $(LCS)_n$-manifolds, Int. J. Pure Appl. Math. 41 (8), 1161-1170, 2007.
[12] A.A. Shaikh, T. Basu and S. Eyasmin, On the existence of ϕ-recurrent $(LCS)_n$- manifolds, Extracta Mathematica 23 (1), 71-83, 2008.
[13] A.A. Shaikh and U.C. De, On 3-dimensional LP-Sasakian manifolds, Soochow J. Math. 26 (4), 359-368, 2000.
[14] A.A. Shaikh and S.K. Hui, On locally $\phi$-symmetric $\beta$-Kenmotsu manifolds, Extracta Mathematica 24 (3), 301-316, 2010.
[15] A.A. Shaikh and S.K. Hui, On extended $\phi$-recurrent $\beta$-Kenmotsu manifolds, Publi. de l’ Inst. Math., Nouvelle serie, 89 (103), 77-88, 2011.
[16] A.A. Shaikh and H. Kundu, On equivalency of various geometric structures, J. Geom. 105, 139-165, 2014.
[17] Z.I. Szabó, Structure theorems on Riemannian spaces satisfying $R(X,Y)\cdot R=0$, I, The local version, J. Diff. Geom. 17, 531-582, 1982.
[18] Z.I. Szabó, Structure theorems on Riemannian spaces satisfying $R(X,Y)\cdot R=0$, II, Global version, Geom. Dedicata 19, 65-108, 1983.
[19] Z.I. Szabó, Classification and construction of complete hypersurfaces satisfying $R(X,Y)\cdot R=0$, Acta. Sci. Math. 47, 321-348, 1984.
[20] T. Takahashi, Sasakian $\phi$-symmetric spaces, Tohoku Math. J. 29, 91-113, 1977.
[21] S. Tanno, Isometric immersions of Sasakian manifold in spheres, Kodai Math. Sem. Rep. 21, 448-458, 1969.
[22] K. Yano and M. Kon, Structures on manifolds, World Scientific Publ., Singapore, 1984.
[23] H. Weyl, Reine infinitesimal geometrie, Math. Zeitschrift 2, 384-411, 1918.

There are 23 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Mathematics
Authors	Absos Ali Shaikh 0000-0001-6312-2564 Chandan Kumar Mondal 0000-0002-3485-9193 Helaluddin Ahmad 0000-0003-2289-9996
Publication Date	August 8, 2019
Published in Issue	Year 2019 Volume: 48 Issue: 4

Cite

APA	Shaikh, A. A., Mondal, C. K., & Ahmad, H. (2019). On locally $\phi$-semisymmetric Sasakian manifolds. Hacettepe Journal of Mathematics and Statistics, 48(4), 1156-1169.
AMA	Shaikh AA, Mondal CK, Ahmad H. On locally $\phi$-semisymmetric Sasakian manifolds. Hacettepe Journal of Mathematics and Statistics. August 2019;48(4):1156-1169.
Chicago	Shaikh, Absos Ali, Chandan Kumar Mondal, and Helaluddin Ahmad. “On Locally $\phi$-Semisymmetric Sasakian Manifolds”. Hacettepe Journal of Mathematics and Statistics 48, no. 4 (August 2019): 1156-69.
EndNote	Shaikh AA, Mondal CK, Ahmad H (August 1, 2019) On locally $\phi$-semisymmetric Sasakian manifolds. Hacettepe Journal of Mathematics and Statistics 48 4 1156–1169.
IEEE	A. A. Shaikh, C. K. Mondal, and H. Ahmad, “On locally $\phi$-semisymmetric Sasakian manifolds”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 4, pp. 1156–1169, 2019.
ISNAD	Shaikh, Absos Ali et al. “On Locally $\phi$-Semisymmetric Sasakian Manifolds”. Hacettepe Journal of Mathematics and Statistics 48/4 (August 2019), 1156-1169.
JAMA	Shaikh AA, Mondal CK, Ahmad H. On locally $\phi$-semisymmetric Sasakian manifolds. Hacettepe Journal of Mathematics and Statistics. 2019;48:1156–1169.
MLA	Shaikh, Absos Ali et al. “On Locally $\phi$-Semisymmetric Sasakian Manifolds”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 4, 2019, pp. 1156-69.
Vancouver	Shaikh AA, Mondal CK, Ahmad H. On locally $\phi$-semisymmetric Sasakian manifolds. Hacettepe Journal of Mathematics and Statistics. 2019;48(4):1156-69.