The Application of A New Type of Connection on $3$-dimensional Quasi-Sasakian Manifolds

Year 2025, Volume: 10 Issue: 2, 71 - 96, 30.08.2025

Azime Çetinkaya

https://doi.org/10.30931/jetas.1699264

Abstract

Firstly we define a new type of quarter symmetric non-metric connection on almost
contact metric manifolds. Using this connection, we investigate some
curvature conditions on $3$-dimensional quasi-Sasakian manifolds e.g. $(\widetilde{R}(X,\xi).\widetilde{R})(Y,V)W=0$, $(\widetilde{P}(X,\xi).\widetilde{H})(Y,V)W=0$, $(\widetilde{R}(X,\xi).\widetilde{S})(Y,Z)=0,$ $(\widetilde{H}(X,\xi).\widetilde{S})(Y,Z)=0,$ and $(\widetilde{P}(X,\xi).\widetilde{S})(Y,Z)=0$. Also we study $\eta $-parallel $3$-dimensional quasi-Sasakian manifolds respect to this connection. Finally we give an example about 3-dimensional quasi-Sasakian manifold given with this connection.

Keywords

Quarter-Symmetric Non-Metric Connection , 3-dimensional quasi-Sasakian manifolds , $\eta $-parallel.

3-boyutlu Quasi-Sasakian Manifoldlarda Tanımlanan Yeni Bir Koneksiyonun Uygulaması

Abstract

Öncelikle, hemen hemen değme metrik manifoldlar üzerinde yeni bir çeyrek simetrik metrik olmayan koneksiyon tanımlayacağız. Bu koneksiyonu kullanarak, 3 boyutlu Quasi-Sasakian manifoldlar üzerinde $(\widetilde{R}(X,\xi).\widetilde{R})(Y,V)W=0$, $(\widetilde{P}(X,\xi).\widetilde{H})(Y,V)W=0$, $(\widetilde{R}(X,\xi).\widetilde{S})(Y,Z)=0,$ $(\widetilde{H}(X,\xi).\widetilde{S})(Y,Z)=0,$ ve $(\widetilde{P}(X,\xi).\widetilde{S})(Y,Z)=0$ gibi bazı eğrilik koşullarını araştıracağız. Ayrıca bu koneksiyona göre $\eta $-paralel olma şartını inceleyeceğiz. Son olarak bu koneksiyon ile verilen 3 boyutlu Quasi-Sasakian manifold hakkında bir örnek oluşturacağız.

Keywords

Quarter-Simetrik Metrik Olmayan Koneksiyon , 3 boyutlu Quasi-Sasakian Manifold , $\eta $-paralel

References

• [1] Agashe, N.S., Chafle, M.R., “A semi-symmetric non-metric connection on a Riemannian manifold”, Indian J. Pure apl. Math. 23(6) (1992) : 399-409.
• [2] Ayar, G., “Some curvature tensor relations on nearly cosymplectic manifolds with Tanaka-Webster connection”, Universal Journal of Mathematics and Applications 5(1) (2022) : 24-31.
• [3] provide new insights into spacetime geometry and gravitationalAyar, G., Aktan, N., Madan, C., “Concircular curvature tensor of nearly cosymplectic manifolds in terms of the generalized Tanaka-Webster connection”, Filomat 38(9) (2024) : 3011-3020.
• [4] Blair, D.E., "Contact manifolds in Riemannian geometry", Lecture Notes in Mathematics 509, Springer-Verlag, Berlin-New York, 1976.
• [5] Blair, D.E., “Riemannian Geometry of Contact and Symplectic Manifolds”, Progress in Mathematics 203, Birkhauser, Boston, 2002.
• [6] Blair, D.E., "The theory of quasi-Sasakian structure", J. Differential Geo. 1 (1967) : 331-345.
• [7] Chaki, M.C., "On pseudo symmetric manifolds", An. Stiint. Univ. Al. I. Cuza Iasi Sect. I a Mat. 33(1) (1987) : 53-58.
• [8] De, U.C., Pathok, G., "On a semi-symmetric metric connection in a Kenmotsu manifold". Bulletin of the Calcutta Mathematical Society 94 (2002) : 319-324.
• [9] De, U.C., Mondal, A.K., "3-dimensional quasi-Sasakian manifolds and Ricci solitons", SUT J. Math. 48(1) (2012) : 71–81.
• [10] De, U.C., Sarkar, A., "On three-dimensional quasi-Sasakian manifolds", SUT J. Math. 45(1) (2009) : 59–71.
• [11] Deprez, J., Roter, W., Verstraelen, L., "Conditions on the projective curvature tensor of conformally flat Riemannian manifolds", Kyungpook Math. J. 29(2) (1989) : 153-166.
• [12] Friedmann, A., Schouten, J.A., "Uber die geometrie der halbsymmetrischen", Ubertragung, Math. Z. 21 (1924) : 211-223.
• [13] Golab, S., "On semi-symmetric and quarter-symmetric linear connections", Tensor 2 (1975) : 249-25.
• [14] Gonzalez, J.C., Chinea, D., "Quasi-Sasakian homogeneous structures on the generalized Heisenberg group H(p,1)", Proc. Amer. Math 105 (1989) : 173-184.
• [15] Hayden, H.A., "Sub-spaces of a space with torsion", Proc. London Math. Soc. 34 (1932) : 27-50.
• [16] Imai, T., "Notes on semi-symmetric metric connections", Tensor (N.S.) 24 (1972) : 293-296.
• [17] Mishra, R.S., Pandey, S.N., "On quarter-symmetric metric connections", Tensor 34 (1980) : 249-254.
• [18] Mishra, R.S., Pandey, S.N., "Semi-symmetric metric connections in an almost contact manifold". Indian Journal of Pure and Applied Mathematics 9(6) (1978) : 570-580.
• [19] Montano, B.C., Nicola, A.D., Dileo, G., "3-quasi-Sasakian manifolds", Annals of Global Analysis and Geometry 33 (2008) : 397-409.
• [20] Kanemaki, S., "Quasi-Sasakian manifolds", Tohoku Math. J. 29 (1977) : 227-233.
• [21] Kanemaki, S., "On quasi-Sasakian manifolds", Differential Geometry Banach center publications 12 (1984) : 95-125.
• [22] Kim, B.H., "Fibred Riemannian spaces with quasi-Sasakian structure", Hiroshima Math. J. 20 (1990) : 477-513.
• [23] Olszak, Z., "Normal almost contact metric manifolds of dimension 3", Ann. Polon. Math. 47 (1986) : 41-50.
• [24] Olszak, Z., Curvature properties of quasi-Sasakian manifolds. Tensor 38 (1982) : 19–28.
• [25] Olszak, Z., "On three dimensional conformally flat quasi-Sasakian manifold", Period Math. Hungar 33 (1996) : 105-112.
• [26] Oubina, J.A., "New classes of almost contact metric structures", Publ. Math. Debrecen 32 (1985) : 187-193.
• [27] Pravanović, M., Pušić, N., "On manifolds admitting some semi-symmetric metric connection", Indian Journal of Mathematics 37(1995) : 37-67.
• [28] Sardar, A., Sarkar, A., "Different solitons associated with 3-dimensional generalized Sasakian-space-forms", Afrika Matematika 36(1) (2025) : 10.
• [29] Sasaki, S., "Almost Contact Manifolds", Part 1, Lecture Notes, Mathematical Institute Tohoku University (1965).
• [30] Sengupta, J., Biswas, B., "Quarter-symmetric non-metric connection on a Sasakian manifolds", Bull. Cal. Math. Soc. 95(2) (2003) : 169-176.
• [31] Tanno, S., "Quasi-Sasakian structure of rank 2p+1", J. Differential Geom. 5 (1971) : 317-324.
• [32] Tripathi, M.M., "A New Connection in a Riemannian manifold", I. Electronic Journal of Geometry 1 (2008) : 15-24.
• [33] Yano, K., "On semi-symmetric metric connection", Rev. Roumaine Math. Pure Appl. 15(9) (1970) : 1579-1586.
• [34] Yano, K., "Concircular Geometry I", Concircular Transformations. Proc Imperial Acad (1940).
• [35] Yano, K., Kon, M., "Structure on manifolds", World Scientific, 1984.
• [36] Yano, K., Imai, T., "Quarter-symmetric metric connections and their curvature tensors", Tensor N. S. 38 (1984) : 13-18.
• [37] Yano, K., Sawaki, S., "Riemannian manifolds admitting a conformal transformation group", J. Differential Geom. 2 (1968) : 161-184.