In particular, the conditions of semi-symmetry, Ricci semi-symmetry, and projective semi-symmetry of this quarter-symmetric metric connection are investigated.
📌 3 Key Takeaways
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Objective & Hypothesis: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.
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Key Findings & Data: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.
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Conclusion & Impact:In particular, the conditions of semi-symmetry, Ricci semi-symmetry, and projective semi-symmetry of this quarter-symmetric metric connection are investigated.
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.
Vanlı, A. T. (2020). Semi-Symmetry Properties of $S$-Manifolds Admitting a Quarter-Symmetric Metric Connection. Hagia Sophia Journal of Geometry, Cilt: 2 Sayı: 2, 38–47. https://ajindex.com/articles/1490457
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.
Blair, D. E. (1971). On a generalization of the Hopf fibration. An. St. Univ. I. Cuza, 17, 171-177.
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.
Deszcz, R. (1992). On pseudosymmetric space. Bull. Soc. Math. Belg., Ser. A, 44, 1-34.
Golab, S. (1975). On semi-symmetric and quarter-symmetric linear connections. Tensor, NS, 29, 249-254.
Goldberg, S. I., & Yano, K. (1970). On normal globally framed f-manifolds. Tohoku Mathematical Journal, Second Series, 22(3), 362-370.
Hasegawa, I., Okuyama, Y., & Abe, T. (1986). On $p$-th Sasakian manifolds. J. Hokkaido Univ. Ed. Sect. II A, 37(1), 1-16.
Hayden, H. A. (1932). Sub‐Spaces of a Space with Torsion. Proceedings of the London Mathematical Society, 2(1), 27-50.
Khan, Q. (2006). On an Einstein projective Sasakian manifold. Novi Sad Journal of Mathematics, 36(1), 97-102.
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.
Perrone, D. (1992). Contact Riemannian manifolds satisfying $R(X,\xi ).R=0$. Yokohama Math. J. 39, 141-149.
Rastogi, S. C. (1978). On quarter-symmetric metric connection. CR Acad. Sci. Bulgar, 31, 811-814.
Rastogi, S. C. (1987). On quarter-symmetric metric connections. Tensor, 44(2), 133-141.
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.
Takahashi, T. (1969). Sasakian manifold with pseudo-Riemannian metric. Tohoku Mathematical Journal, Second Series, 21(2), 271-290.
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.
Yano, K. (1970). On semi-symmetric connection. Revue Roumaine de Math. Pure et Appliques, 15, 1570-1586.