Semi-Symmetry Properties of $S$-Manifolds Admitting a Quarter-Symmetric Metric Connection

Year 2020, Volume: 2 Issue: 2, 38 - 47, 09.12.2020

Aysel Turgut Vanlı

Abstract

In this study $S$-manifolds admitting a quarter-symmetric metric connection naturally related with the $S$-structure are considered and some general results concerning the curvature of such a connection is given. In addition, we prove that an $S$-manifold has constant $f$-sectional curvature with respect to this quarter-symmetric metric connection if and only if has the same constant $f$-sectional curvature with respect to the Riemannian connection. In particular, the conditions of semi-symmetry, Ricci semi-symmetry, and projective semi-symmetry of this quarter-symmetric metric connection are investigated.

Keywords

$S$-manifold , quarter-symmetric metric connection , semi-symmetry properties

References

• [1] Blair, D. E. (1970). Geometry of manifolds with structural group ${\mathcal U}(n)\times {\mathcal O}(s)$. Journal of differential geometry, 4(2), 155-167.
• [2] Blair, D. E. (1971). On a generalization of the Hopf fibration. An. St. Univ. I. Cuza, 17, 171-177.
• [3] Cabrerizo, J. L., Fernandez, L. M., & Fernandez, M. (1990). The curvature tensor fields on f-manifolds with complemented frames. An. Stiint. Univ. Al. I. Cuza Iasi, 36, 151-161.
• [4] Deszcz, R. (1992). On pseudosymmetric space. Bull. Soc. Math. Belg., Ser. A, 44, 1-34.
• [5] Golab, S. (1975). On semi-symmetric and quarter-symmetric linear connections. Tensor, NS, 29, 249-254.
• [6] Goldberg, S. I., & Yano, K. (1970). On normal globally framed f-manifolds. Tohoku Mathematical Journal, Second Series, 22(3), 362-370.
• [7] Hasegawa, I., Okuyama, Y., & Abe, T. (1986). On $p$-th Sasakian manifolds. J. Hokkaido Univ. Ed. Sect. II A, 37(1), 1-16.
• [8] Hayden, H. A. (1932). Sub‐Spaces of a Space with Torsion. Proceedings of the London Mathematical Society, 2(1), 27-50.
• [9] Khan, Q. (2006). On an Einstein projective Sasakian manifold. Novi Sad Journal of Mathematics, 36(1), 97-102.
• [10] Kobayashi, M., & Tsuchiya, S. (1972). Invariant submanifolds of an f-manifold with complemented frames. In Kodai Mathematical Seminar Reports, Vol. 24, No. 4, pp. 430-450. Department of Mathematics, Tokyo Institute of Technology.
• [11] Perrone, D. (1992). Contact Riemannian manifolds satisfying $R(X,\xi ).R=0$. Yokohama Math. J. 39, 141-149.
• [12] Rastogi, S. C. (1978). On quarter-symmetric metric connection. CR Acad. Sci. Bulgar, 31, 811-814.
• [13] Rastogi, S. C. (1987). On quarter-symmetric metric connections. Tensor, 44(2), 133-141.
• [14] Szabó, Z. I. (1982). Structure theorems on Riemannian spaces satisfying $R(X,\,Y)\cdot R=0$. I. The local version. Journal of Differential Geometry, 17(4), 531--582. doi:10.4310/jdg/1214437486.
• [15] Takahashi, T. (1969). Sasakian manifold with pseudo-Riemannian metric. Tohoku Mathematical Journal, Second Series, 21(2), 271-290.
• [16] Yano, K. (1963). On structure defined by a tensor field $f$ of type (1,1) satisfying $f^{3}+f=0$. Tensor, NS, 14, 99-109.
• [17] Yano, K. (1970). On semi-symmetric connection. Revue Roumaine de Math. Pure et Appliques, 15, 1570-1586.