Generalized η-Ricci solitons on f-Kenmotsu 3-manifolds associated to the Schoutenvan Kampen connection

Document Type : Original Article


Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran


In this paper, we investigate f-Kenmotsu 3-dimensional manifolds admitting generalized η-Ricci solitons with respect to the Schouten-van Kampen connection. We provide an example of generalized η-Ricci solitons with respect to the Schouten-van Kampen connection on an f-Kenmotsu 3-dimensional manifold to prove our results.


Main Subjects

[1] S. Azami, Generalized Ricci solitons of three-dimensional Lorentzian Lie groups associated canonical connec[1]tions and Kobayashi-Nomizu connections, J. Nonlinear Math. Phys., 30 (2023), pp. 1–33.
[2] A. Bejancu, Schouten-Van Kampen and Vr˘anceanu connections on foliated manifolds, An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si. Mat. (N.S.), 52 (2006), pp. 37–60.
[3] A. M. Blaga, Canonical connections on para-Kenmotsu manifolds, Novi Sad J. Math., 45 (2015), pp. 131–142. [4] , η-Ricci solitons on para-Kenmotsu manifolds, Balkan J. Geom. Appl., 20 (2015), pp. 1–13.
[5] , η-Ricci solitons on Lorentzian para-Sasakian manifolds, Filomat, 30 (2016), pp. 489–496.
[6] A. M. Blaga and M. Crasmareanu, Torse-forming η-Ricci solitons in almost paracontact η-Einstein geometry, Filomat, 31 (2017), pp. 499–504.
[7] D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, Vol. 509, SpringerVerlag, Berlin-New York, 1976.
[8] , Riemannian geometry of contact and symplectic manifolds, vol. 203 of Progress in Mathematics, Birkh¨auser Boston, Inc., Boston, MA, 2002.
[9] G. Calvaruso, Three-dimensional homogeneous generalized Ricci solitons, Mediterr. J. Math., 14, 216 (2017).
[10] T. Chave and G. Valent, On a class of compact and non-compact quasi-Einstein metrics and their renor[1]malizability properties, Nuclear Phys. B, 478 (1996), pp. 758–778.
[11] , Quasi-Einstein metrics and their renormalizability properties, Helv. Phys. Acta. 69 (1996), pp. 344–347.
[12] B.-Y. Chen, A simple characterization of generalized Robertson–Walker spacetimes, Gen. Rel. Grav., 46 (2014), pp. 1–5.
[13] , Classification of torqued vector fields and its applications to Ricci solitons, Kragujevac J. Math., 41 (2017), pp. 239–250.
[14] J. T. Cho and M. Kimura, Ricci solitons and real hypersurfaces in a complex space form, Tohoku Math. J. (2), 61 (2009), pp. 205–212.
[15] C. Calin and M. Crasmareanu ˘ , From the Eisenhart problem to Ricci solitons in f-Kenmotsu manifolds, Bull. Malays. Math. Sci. Soc. (2), 33 (2010), pp. 361–368.
[16] G. Ghosh, On Schouten–van Kampen connection in Sasakian manifolds, Bol. Soc. Parana. Mat. (3), 36 (2018), pp. 171–182.
[17] R. S. Hamilton, The Ricci flow on surfaces, in Mathematics and general relativity (Santa Cruz, CA, 1986), vol. 71 of Contemp. Math., Amer. Math. Soc., Providence, RI, 1988, pp. 237–262.
[18] A. Haseeb, S. Pandey, and R. Prasad, Some results on η-Ricci solitons in quasi-Sasakian 3-manifolds, Commun. Korean Math. Soc., 36 (2021), pp. 377–387.
[19] A. Kazan and H. B. Karadag, Trans-sasakian manifolds with Schouten-Van Kampen connection, Ilirias J. Math., 7 (2018), pp. 1–12.
[20] K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. J. (2), 24 (1972), pp. 93–103. [21] D. L. Kiran Kumar, H. G. Nagaraja, and S. H. Naveenkumar, Some curvature properties of Kenmotsu manifolds with Schouten–van Kampen connection, Bull. Transilv. Univ. Bra¸sov Ser. III, 12(61) (2019), pp. 351– 364.
[22] P. Majhi, U. C. De, and D. Kar, η-Ricci solitons on Sasakian 3-manifolds, An. Univ. Vest Timi¸s. Ser. Mat.-Inform., 55 (2017), pp. 143–156.
[23] M. E. A. Mekki and A. M. Cherif, Generalised Ricci solitons on Sasakian manifolds, Kyungpook Math. J., 57 (2017), pp. 677–682.
[24] P. Nurowski and M. Randall, Generalized Ricci solitons, J. Geom. Anal., 26 (2016), pp. 1280–1345.
[25] Z. Olszak, Locally conformal almost cosymplectic manifolds, Colloq. Math., 57 (1989), pp. 73–87.
[26] , The Schouten-van Kampen affine connection adapted to an almost (para) contact metric structure, Publ. Inst. Math. (Beograd) (N.S.), 94(108) (2013), pp. 31–42.
[27] Z. Olszak and R. Ros¸ca, Normal locally conformal almost cosymplectic manifolds, Publ. Math. Debrecen, 39 (1991), pp. 315–323.
[28] S. Y. Perktas¸ and A. Yı ldız, On quasi-Sasakian 3-manifolds with respect to the Schouten–van Kampen connection, Int. Electron. J. Geom., 13 (2020), pp. 62–74.
[29] D. G. Prakasha and B. S. Hadimani, η-Ricci solitons on para-Sasakian manifolds, J. Geom., 108 (2017), pp. 383–392.
[30] J. A. Schouten, Ricci-calculus. An introduction to tensor analysis and its geometrical applications, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Ber¨ucksichtigung der Anwendungsgebiete, Band X, Springer-Verlag, Berlin-G¨ottingen-Heidelberg, 1954. 2d. ed.
[31] J. A. Schouten and E. R. van Kampen, Zur Einbettungs- und Kr¨ummungstheorie nichtholonomer Gebilde, Math. Ann., 103 (1930), pp. 752–783.
[32] M. D. Siddiqi, Generalized η-Ricci solitons in trans Sasakian manifolds, Eurasian Bull. Math., 1 (2018), pp. 107–116.
[33] A. F. Solov’ev, On the curvature of the connection induced on a hyperdistribution in a Riemannian space, Geom. Sb., 19 (1978), pp. 12–23.
[34] , The bending of hyperdistributions, Geom. Sb., 20 (1979), pp. 101–112.
[35] , Second fundamental form of a distribution, Math. Notes, 31 (1982), pp. 71–75.
[36] M. Turan, C. Yetim, and S. K. Chaubey, On quasi-Sasakian 3-manifolds admitting η-Ricci solitons, Filomat, 33 (2019), pp. 4923–4930.
[37] L. Vanhecke and D. Janssens, Almost contact structures and curvature tensors, Kodai Math. J., 4 (1981), pp. 1–27.
 [38] G. Vranceanu ˇ , Sur quelques points de la th´eorie des espaces non-holonomes, Bull. Fac. S¸t. Cernaut¸i 5 (1931), pp. 177–205.
[39] K. Yano, Concircular geometry. I. Concircular transformations, Proc. Imp. Acad. Tokyo, 16 (1940), pp. 195– 200.
[40] , On the torse-forming directions in Riemannian spaces, Proc. Imp. Acad. Tokyo, 20 (1944), pp. 340–345.
[41] K. Yano and B.-y. Chen, On the concurrent vector fields of immersed manifolds, K¯odai Math. Sem. Rep., 23 (1971), pp. 343–350.
[42] S. Yuksel Perktas¸ and A. Yildiz ¨ , On f-Kenmotsu 3-manifolds with respect to the Schouten–van Kampen connection, Turkish J. Math., 45 (2021), pp. 387–409.