On Quasi-Sasakian $3$-Manifolds with Respect to the Schouten-van Kampen Connection

Year 2020, Volume: 13 Issue: 2, 62 - 74, 15.10.2020

Selcen Yüksel Perktaş Ahmet Yıldız

https://doi.org/10.36890/iejg.742073

Cited By: 4

Abstract

In this paper we study some soliton types on a quasi-Sasakian 3-manifold with respect to the Schouten-van Kampen connection.                                                                                                                                             ..............................................                                                                                                                 .................................................................................................................................................................................................................................................................................................       

Keywords

Quasi-Sasakian manifolds, Schouten-van Kampen connection, Ricci soliton, \eta-Ricci soliton, Yamabe soliton

References

• [1] Barbosa E., Riberio E.: On conformal solutions of the Yamabe flow. Arch. Math. 101, 79-89 (2013).
• [2] Blair D. E.: Contact manifolds in Riemannian geometry. Lecture Notes in Mathematics Vol 509. Springer-Verlag, Berlin-New York (1976).
• [3] Blair D. E.: Riemannian geometry of contact and symplectic manifolds. Progress in Mathematics Vol 203. Birkhauser Boston Inc. (2002).
• [4] Blair D. E.: The theory of quasi-Sasakian structure. J. Differential Geo. 1, 331-345 (1967).
• [5] Bejancu A., Faran H.: Foliations and geometric structures. Math. and Its Appl. 580. Springer, Dordrecht (2006).
• [6] Cao H. D., X. Sun X., Zhang Y.: On the structure of gradient Yamabe solitons. Mathematical Research Letters. 19, 767-774 (2012).
• [7] Chen B. Y., Deshmukh S.: Yamabe and quasi Yamabe solitons on Euclidean submanifolds. Mediterr. J. Math. 15(5), 1-9 (2018).
• [8] Cho J. C., Kimura M.: Ricci solitons and real hypersurfaces in a complex space form. Tohoku Math. J. 61(2), 205-212 (2009).
• [9] Chow B., Knopf D.: The Ricci flow: An introduction, Mathematical Surveys and Monographs 110. American Math. Soc. (2004).
• [10] De U. C., Yıldız A., Sarkar A.: Isometric immersion of three-dimensional quasi-Sasakian manifolds. Math. Balkanica. 22(3-4), 297-306 (2008).
• [11] De U. C., Mandal A. K.: 3-dimensional quasi-Sasakian manifolds and Ricci solitons, Sut Journal of Mathematics. 48(1), 71-81 (2012).
• [12] Derdzinski A.: Compact Ricci solitons. preprint.
• [13] Deshmukh S., Chen B. Y.: A note on Yamabe solitons. Balkan J. Geom. Appl. 23(1), 37-43 (2018).
• [14] Gonzalez J. C. , Chinea D.: Quasi-Sasakian homogeneous structures on the generalized Heisenberg group H(p,1). Proc. Amer. Math. 105, 173-184 (1989).
• [15] Hamilton R. S.: The Ricci flow on surfaces. Mathematics and general relativity, Contemp. Math. 71, 237-262 (1988).
• [16] Ianu¸s S.: Some almost product structures on manifolds with linear connection. Kodai Math. Sem. Rep. 23, 305-310 (1971).
• [17] Ivey T.: Ricci solitons on compact 3-manifolds. Differential Geo. Appl. 3, 301-307 (1993).
• [18] Janssens D., Vanhecke L.: Almost contact structures and curvature tensors. Kodai Math. J. 4(1), 1-27 (1981).
• [19] Kim B. H.: Fibred Riemannian spaces with quasi-Sasakian structure. Hiroshima Math. J. 20, 477-513 (1990).
• [20] Kanemaki S.: Quasi-Sasakian manifolds. Tohoku Math. J. 29, 227-233 (1977).
• [21] Kanemaki S.: On quasi-Sasakian manifolds. Differential Geometry Banach Center Publications. 12, 95-125 (1984).
• [22] Neto, B. L.: A note on (anti-)self dual quasi Yamabe gradient solitons. Results Math. 71, 527-533 (2017).
• [23] Olszak Z.: Normal almost contact metric manifolds of dimension 3. Ann. Polon. Math. 47, 41-50 (1986).
• [24] Olszak Z.: On three dimensional conformally flat quasi-Sasakian manifold. Period Math. Hungar. 33(2), 105-113 (1996).
• [25] Olszak Z.: The Schouten-van Kampen affine connection adapted an almost (para) contact metric structure. Publ. De L’inst. Math. 94, 31-42 (2013).
• [26] Oubina J. A.: New classes of almost contact metric structures. Publ. Math. Debrecen. 32, 187-193 (1985).
• [27] Perelman G.: The entopy formula for the Ricci flow and its geometric applications, Preprint arxiv:0211159 (2002).
• [28] Schouten J., van Kampen E.: Zur Einbettungs-und Krümmungsthorie nichtholonomer Gebilde. Math. Ann. 103, 752-783 (1930).
• [29] Sharma R.: Certain results on K-contact and $(k,\sigma )$- contact manifolds. Journal of Geometry. 89, 138-147 (2008).
• [30] Solov’ev A. F.: On the curvature of the connection induced on a hyperdistribution in a Riemannian space. Geom. Sb. 19, 12-23 (1978).
• [31] Solov’ev A. F.: The bending of hyperdistributions. Geom. Sb., 20(1979), 101-112, (in Russian).
• [32] Solov’ev A. F.: Second fundamental form of a distribution. Mat. Zametki, 35, 139-146 (1982).
• [33] Solov’ev A. F.: Curvature of a distribution. Mat. Zametki. 35, 111-124 (1984).
• [34] Tanno S.: Quasi-Sasakian structure of rank 2p+1. J. Differential Geom.