Some results on paracontact metric $(k,\mu)$-manifolds with respect to the Schouten-van Kampen connection

Year 2022, Volume: 51 Issue: 2, 466 - 482, 01.04.2022

Selcen Yüksel Perktaş , U.c. De , Ahmet Yıldız

https://doi.org/10.15672/hujms.941744

Abstract

In the present paper we study certain symmetry conditions and some types of solitons on paracontact metric $(k,\mu )$-manifolds with respect to the Schouten-van Kampen connection. We prove that a Ricci semisymmetric paracontact metric $(k,\mu )$-manifold with respect to the Schouten-van Kampen connection is an $\eta $-Einstein manifold. We investigate paracontact metric $(k,\mu )$-manifolds satisfying $\breve{Q}\cdot \breve{R}_{cur}=0$\ with respect to the Schouten-van Kampen connection. Also, we show that there does not exist an almost Ricci soliton in a $(2n+1)$-dimensional paracontact metric $(k,\mu )$-manifold with respect to the Schouten-van Kampen connection such that $k>-1$ or $k<-1$. In case of the metric is being an almost gradient Ricci soliton with respect to the Schouten-van Kampen connection, then we state that the manifold is either $N(k)$-paracontact metric manifold or an Einstein manifold. Finally, we present some results related to almost Yamabe solitons in a paracontact metric $(k,\mu )$-manifold equipped with the Schouten-van Kampen connection and construct an example which verifies some of our results.

Keywords

Schouten-van Kampen connection, Ricci semisymmetric, Einstein manifold, $\eta $-Einstein manifold, solitons, paracontact metric $(k,\mu )$-manifolds

References

• [1] E. Barbosa and E. Riberio, On conformal solutions of the Yamabe flow, Arch. Math. 101, 79-89, 2013.
• [2] A. Bejancu and H. Faran, Foliations and geometric structures, Math. and Its Appl. 580, Springer, Dordrecht, 2006.
• [3] A.M. Blaga and S. Yüksel Perktaş, Remarks on almost $\eta$-Ricci solitons in $(\varepsilon)$-para Sasakian manifolds, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68 (2), 1621- 1628, 2019.
• [4] A.M. Blaga, S. Yüksel Perktaş, B. Acet and F.E. Erdoğan, $\eta$-Ricci solitons in $(\varepsilon)$- almost paracontact metric manifolds, Glasnik Matematiki 53 (1), 205-220, 2019.
• [5] H.D. Cao, X. Sun and Y. Zhang, On the structure of gradient Yamabe solitons, Math. Res. Lett. 19, 767-774, 2012.
• [6] B. Cappelletti-Montano, I. Küpeli Erken and C. Murathan, Nullity conditions in paracontact geometry, Diff. Geom. Appl. 30, 665-693, 2012.
• [7] B.Y. Chen and S. Deshmukh, Yamabe and quasi Yamabe solitons on Euclidean submanifolds, Mediterr. J. Math. 15 (5), 194, 1-9, 2018.
• [8] J.T. Cho and M. Kimura, Ricci solitons and real hypersurfaces in a complex space form, Tohoku Math. J. 61 (2), 205-212, 2009.
• [9] B. Chow and D. Knopf, The Ricci flow: An introduction, Mathematical Surveys and Monographs 110, American Math. Soc., 2004.
• [10] A. Derdzinski, Compact Ricci solitons, preprint.
• [11] S. Deshmukh and B.Y. Chen, A note on Yamabe solitons, Balkan J. Geom. Appl. 23 (1), 37-43, 2018.
• [12] R.S. Hamilton, The Ricci flow on surfaces, Mathematics and General Relativity (Santa Cruz, CA, 1986), 237-262, Contemp. Math. 71, American Math. Soc., 1988.
• [13] S. Ianuş, Some almost product structures on manifolds with linear connection, Kodai Math. Sem. Rep. 23, 305-310, 1971.
• [14] T. Ivey, Ricci solitons on compact 3-manifolds, Diff. Geo. Appl. 3, 301-307, 1993.
• [15] S. Kaneyuki and F.L. Williams, Almost paracontact and parahodge structures on manifolds, Nagoya Math. J. 99, 173-187, 1985.
• [16] K. Mandal and U.C. De, Paracontact metric(k, μ)-spaces satisfying certain curvature conditions, Kyungpook Math. J. 59 (1), 163-174, 2019.
• [17] V.A. Mirzoyan, Structure theorems on Riemannian Ricci semisymmetric spaces, Izv. Uchebn. Zaved. Mat. 6, 80-89, 1992.
• [18] B.L. Neto, A note on (anti-)self dual quasi Yamabe gradient solitons, Results Math. 71, 527-533, 2017.
• [19] Z. Olszak, The Schouten-van Kampen affine connection adapted an almost (para) contact metric structure, Publ. De L’inst. Math. 94, 31-42, 2013.
• [20] G. Perelman, The entopy formula for the Ricci flow and its geometric applications, preprint, http://arxiv.org/abs/math.DG/02111159.
• [21] J. Schouten and E. van Kampen, Zur Einbettungs-und Krümmungsthorie nichtholonomer Gebilde, Math. Ann. 103, 752-783, 1930.
• [22] R. Sharma, Certain results on K-contact and ${(k,\mu )}$-contact manifolds, J. Geom. 89, 138-147, 2008.
• [23] A.F. Solov’ev, On the curvature of the connection induced on a hyperdistribution in a Riemannian space, Geom. Sb. 19, 12-23, 1978 (in Russian).
• [24] A.F. Solov’ev, The bending of hyperdistributions, Geom. Sb. 20, 101-112, 1979 (in Russian).
• [25] A.F. Solov’ev, Second fundamental form of a distribution, Mat. Zametki 35, 139-146, 1982.
• [26] A.F. Solov’ev, Curvature of a distribution, Mat. Zametki 35, 111-124, 1984.
• [27] Z. Szabo, Structure theorems on Riemannian spaces satisfying R(X,Y).R=0, The local version, J. Differ. Geom. 17, 531-582, 1982.
• [28] Y. Wang, Yamabe solitons on three-dimensional Kenmotsu manifolds, Bull. Belg. Math. Soc. Simon Stevin 23, 345355, 2016.
• [29] Y. Wang, Ricci solitons on 3-dimensional cosymplectic manifolds, Math. Slovaca 67, 979984, 2017.
• [30] Y. Wang, Almost Kenmotsu ${(k,\mu )}^{'}$-manifolds with Yamabe solitons, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 115, 14-21, 2021.
• [31] Y. Wang and W. Wang, Some results on $(k, μ)$-almost Kenmotsu manifolds, Quaest. Math. 41, 469481, 2018.
• [32] S. Zamkovoy, Canonical connections on paracontact manifolds, Ann. Glob. Anal. Geom. 36, 37-60, 2009.
• [33] S. Zamkovoy and V. Tzanov, Non-existence of flat paracontact metric structures in dimension greater than or equal to five, Annuaire Univ. Sofia Fac. Math. Inform. 100, 27-34, 2011.