$Q$-Curvature Tensor on $f$-Kenmotsu $3$-Manifolds

Year 2022, Volume: 5 Issue: 3, 96 - 106, 30.09.2022

Sunil Yadav Ahmet Yıldız

https://doi.org/10.32323/ujma.1055272

Abstract

The object of the present paper is to consider $f$-Kenmotsu $3$-manifolds fulfilling certain curvature conditions on $Q$-curvature tensor with the Schouten-van Kampen connection. Certain consequences of $Q$-curvature tensor on such manifolds bearing Ricci soliton in perspective of Schouten-van Kampen association are likewise displayed. In the last segment, examples are given.

Keywords

f-Kenmotsu 3-manifolds, Q-curvatute tensor, Schouten-van Kampen connection

References

• [1] D. E. Blair, Contact manifolds in Riemannian geometry, Lect. Notes Math., 509 (1976).
• [2] S. Sasaki, Y. Hatakeyama, On differentiable manifolds with certain structures which are closely related to almost contact structures II, Tohoku Math. J., 13 (1961), 281-294.
• [3] Z. Olszak, R. Rosca, Normal locally conformal almost cosymplectic manifolds, Publ. Math. Debrecen, 39 (1991), 315-323.
• [4] S. Ianus, Some almost product structures on manifolds with linear connection, Kodai Math. Sem. Rep., 23 (1971), 305-310.
• [5] A. Bejancu, H. Faran, Foliations and Geometric Structures, Math. and Its Appl., 580, Springer, Dordrecht, 2006.
• [6] A. F. Solov’ev, On the curvature of the connection induced on a hyperdistribution in a Riemannian space, Geom. Sb., 19 12-23, (1978).
• [7] A. F. Solov’ev, The bending of hyperdistributions, Geom. Sb., 20 (1979), 101-112.
• [8] Z. Olszak, The Schouten-van Kampen affine connection adapted an almost (para) contact metric structure, Publ. De L’inst. Math., 94 (2013), 31-42.
• [9] K. Yano, S. Bochner, Curvature and Betti numbers, Ann. Math. Stud., 32 (1953).
• [10] G. Zhen, J. L. Cabrerizo, L. M. Fern´andez, M. Fern´andez, On x -conformally flat contact metric manifolds, Indian J. Pure Appl. Math., 28, (1997), 725-734.
• [11] A. Yıldız, U. C. De, M. Turan, On 3-dimensional f -Kenmotsu manifolds and Ricci solitons, Ukrainian. Math. J., 65(5) (2013), 620-628.
• [12] U. C. De, A. Yıldız, Certain curvature conditions on generalized Sasakian-space-forms, Quaest. Math., 38(4) (2015), 495-504.
• [13] W. Kuhnel, Conformal transformations between Einstein spaces, In: Conformal Geometry, Vieweg Teubner Verlag, Wiesbaden, 105-146.
• [14] K. Yano, Concircular geometry I. Concircular transformation, Proc. Imp. Acad. Tokyo, 16, (1940), 195-200.
• [15] C. A. Mantica, Y. J. Suh, Pseudo-Q-symmetric Riemannian manifolds, Int. J. Geom. Methods Mod. Phys. 10(5) (2013), 25 pages.
• [16] R. S. Hamilton, The Ricci flow on surfaces, Contemp. Math., 71 (1988), 237-262.
• [17] B. Chow, D. Knopf, The Ricci flow: An introduction, Math. Surv. and Monogram, 110 (2004).
• [18] C. C˘alin, C. Crasmareanu, From the Eisenhart problem to the Ricci solitons in f -Kenmotau manifolds, Bull. Malays. Math. Sci. Soc. (2), 33(3), (2010), 361-368.
• [19] T. Ivey, Ricci solitons on compact 3-manifolds, Different. Geom. Appl., 3 (1993), 301-307.
• [20] A. Derdzinski, A Myers-type theorem and compact Ricci solitons, Proc. Am. Math. Soc., 134(12) (2006), 3645-3648.
• [21] D. Jannsens, L. Vanhecke, Almost contact structures and curvature tensor, Kodai Math. J., 4(1) (1981), 1-27.
• [22] K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. J., 24(1) (1972), 93-103.
• [23] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159v1 [math.DG], (2002).
• [24] A. Yıldız, On f -Kenmotsu manifolds with the Schouten-van Kampen connection, Publ. de l’Institut Math., Nouvelle s´erie, tome 102(116) (2017), 93-105.
• [25] S. Y. Perktas¸, A. Yıldız, On f -Kenmotsu 3-manifolds with respect to the Schouten-van Kampen connection, Turk. J. Math., 45 (2021), 387-409.