Sasakian 3-manifolds admitting a gradient Ricci-Yamabe soliton
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Abstract
The object of the present paper is to characterize Sasakian 3-manifolds admitting a gradient Ricci-Yamabe soliton. It is shown that a Sasakian 3-manifold $M$ with constant scalar curvature admitting a proper gradient Ricci-Yamabe soliton is Einstein and locally isometric to a unit sphere. Also, the potential vector field is an infinitesimal automorphism of the contact metric structure. In addition, if $M$ is complete, then it is compact.
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[1] Y. Akrami, T. S. Koivisto and A. R. Solomon, The nature of spacetime in bigravity: two metrics or none?, Gen. Relativ. and Grav. 47 (2015), 1838. Google Scholar
[2] C. P. Boyer and K. Galicki, On Sasakian Einstein geometry, Int. J. Math. 11 (2000), 873–909. Google Scholar
[3] C. P . Boyer, K. Galicki and P. Matzeu, On η-Einstein Sasakian geometry, Comm. Math. Phys. 262 (2006), 177–208. Google Scholar
[4] W. Boskoff and M. Crasmareanu, A Rosen type bi-metric universe and its physical properties, Int. J. Geom. Methods Mod. Phys. 15 (2018), 1850174. Google Scholar
[5] P. Candelas, G. T. Horowitz, A. Strominger and E. Witten, Vacuum configuration for super- strings, Nuclear Phys. B 258 (1995), 46–74. Google Scholar
[6] G. Catino and L. Mazzieri, Gradient Einstein solitons, Nonlinear Anal. 132 (2016), 66–94. Google Scholar
[7] G. Catino, L. Cremaschi, Z. Djadli, C. Mantegazza and L. Mazzieri, The Ricci-Bourguignon flow, Pacific J. Math. 28 (2017), 337–370. Google Scholar
[8] D. Dey, Almost Kenmotsu metric as Ricci-Yamabe soliton, arXiv:2005.02322v1 [math.DG] Google Scholar
[9] A. Ghosh and R. Sharma, Sasakian metric as a Ricci soliton and related results, J. Geom. Phys. 75 (2014), 1–6. Google Scholar
[10] S. Gu ̈ler and M. Crasmareanu, Ricci-Yamabe maps for Riemannian flows and their volume variation and volume entropy, Turk. J. Math. 43 (2019), 2361–2641. Google Scholar
[11] R. S. Hamilton, Three manifold with positive Ricci curvature, J. Differential Geom. 17 (1982), 255–306. Google Scholar
[12] R. S. Hamilton, Ricci flow on surfaces, Contempory Mathematics 71 (1988), 237–261. Google Scholar
[13] R. S. Hamilton, Lectures on geometric flows, 1989 (unpublished). Google Scholar
[14] J. B. Jun and U. K. Kim, On 3-dimensional almost contact metric manifolds, Kyungpook Math. J. 34 (1994), 293–301. Google Scholar
[15] P. Majhi, U. C. De and D. Kar, η-Ricci solitons on Sasakian 3-manifolds, An. Univ. Vest Timis. Ser. Mat.-Inform. LV, 2 (2017), 143–156. Google Scholar
[16] J. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theoret. Phys. 38 (1999), 1113–1133. Google Scholar
[17] S. B. Myers, Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941), 401–404. Google Scholar
[18] R. Sharma, Certain results on K-contact and (k,μ)-contact manifolds, J. Geom. 89 (2008), 138–147. Google Scholar
[19] R. Sharma, A 3-dimensional Sasakian metric as a Yamabe soliton, Int. J. Geom. Methods Mod. Phys. 09 (2012), 1220003. Google Scholar
[20] S. Sasaki, Lecture notes on almost contact manifolds-I, Mathematical Institute, Tohoku Univ.(1965) Google Scholar
[21] M. D. Siddiqi and M. A. Akyol, η-Ricci-Yamabe solitons on Riemannian submersions from Riemannian manifolds, arXiv.2004.14124v1 [math.DG] Google Scholar
[22] S. Tanno, Note on infinitesimal transformations over contact manifolds, Tohoku Math. J. 14 (1962), 416–430. Google Scholar
[23] V. Venkatesha and D. M. Naik, Yamabe solitons on 3-dimensional contact metric manifolds with Qφ = φQ, Int. J. Geom. Methods Mod. Phys. 16 (2019), 1950039. Google Scholar