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

Year 2020, , 62 - 74, 15.10.2020
https://doi.org/10.36890/iejg.742073

Abstract

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

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.
Year 2020, , 62 - 74, 15.10.2020
https://doi.org/10.36890/iejg.742073

Abstract

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.
There are 34 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Selcen Yüksel Perktaş 0000-0002-8848-0621

Ahmet Yıldız 0000-0002-9799-1781

Publication Date October 15, 2020
Acceptance Date June 9, 2020
Published in Issue Year 2020

Cite

APA Yüksel Perktaş, S., & Yıldız, A. (2020). On Quasi-Sasakian $3$-Manifolds with Respect to the Schouten-van Kampen Connection. International Electronic Journal of Geometry, 13(2), 62-74. https://doi.org/10.36890/iejg.742073
AMA Yüksel Perktaş S, Yıldız A. On Quasi-Sasakian $3$-Manifolds with Respect to the Schouten-van Kampen Connection. Int. Electron. J. Geom. October 2020;13(2):62-74. doi:10.36890/iejg.742073
Chicago Yüksel Perktaş, Selcen, and Ahmet Yıldız. “On Quasi-Sasakian $3$-Manifolds With Respect to the Schouten-Van Kampen Connection”. International Electronic Journal of Geometry 13, no. 2 (October 2020): 62-74. https://doi.org/10.36890/iejg.742073.
EndNote Yüksel Perktaş S, Yıldız A (October 1, 2020) On Quasi-Sasakian $3$-Manifolds with Respect to the Schouten-van Kampen Connection. International Electronic Journal of Geometry 13 2 62–74.
IEEE S. Yüksel Perktaş and A. Yıldız, “On Quasi-Sasakian $3$-Manifolds with Respect to the Schouten-van Kampen Connection”, Int. Electron. J. Geom., vol. 13, no. 2, pp. 62–74, 2020, doi: 10.36890/iejg.742073.
ISNAD Yüksel Perktaş, Selcen - Yıldız, Ahmet. “On Quasi-Sasakian $3$-Manifolds With Respect to the Schouten-Van Kampen Connection”. International Electronic Journal of Geometry 13/2 (October 2020), 62-74. https://doi.org/10.36890/iejg.742073.
JAMA Yüksel Perktaş S, Yıldız A. On Quasi-Sasakian $3$-Manifolds with Respect to the Schouten-van Kampen Connection. Int. Electron. J. Geom. 2020;13:62–74.
MLA Yüksel Perktaş, Selcen and Ahmet Yıldız. “On Quasi-Sasakian $3$-Manifolds With Respect to the Schouten-Van Kampen Connection”. International Electronic Journal of Geometry, vol. 13, no. 2, 2020, pp. 62-74, doi:10.36890/iejg.742073.
Vancouver Yüksel Perktaş S, Yıldız A. On Quasi-Sasakian $3$-Manifolds with Respect to the Schouten-van Kampen Connection. Int. Electron. J. Geom. 2020;13(2):62-74.