Abstract
This paper aims to classify a certain type of three-dimensional complete non-Sasakian contact manifold with specific properties, namely \(Q\xi =\sigma \xi \) and admitting Ricci–Bourguignon solitons. In the case of constant \(\sigma \), the paper proves that if the potential vector field of the Ricci–Bourguignon soliton is orthogonal to the Reeb vector field, then the manifold is either Einstein or locally isometric to E(1, 1). Under a similar hypothesis, the paper shows that a \((\kappa ,\mu ,\vartheta )\)-contact metric manifold is locally isometric to E(1, 1). Finally, the paper considers the scenario where the potential vector is pointwise collinear with the Reeb vector field and presents some results.
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References
Bourguignon, J.P.: Ricci curvature and Einstein metrics, In: Global Differential Geometry and Global Analysis (Berlin, 1979), Lecture Notes in Mathemathics, vol. 838, pp. 42–63 (1981)
Catino, G., Cremaschi, L., Djadli, Z., Mantegazza, C., Mazzieri, L.: The Ricci-Bourguignon flow. Pac. J. Math. 2, 337–370 (2017)
Wang, L.F.: Monotonicity of eigenvalues and functionals along the Ricci–Bourguignon flow. J. Geom. Anal. 29, 1116–1135 (2019)
Khatri, M., Zosangzuala, C., Singh, J.P.: Isometries on almost Ricci–Yamabe solitons. Arab. J. Math. 12, 127–138 (2023)
Singh, J.P., Khatri, M.: On Ricci–Yamabe Soliton and geometrical structure in a perfect fluid spacetime. Afr. Mat. 32(5), 1645–1656 (2021)
Dwivedi, S.: Some results on Ricci–Bourguignon Solitons and almost solitons. Can. Math. Bull. 64(3), 591–604 (2021)
Catino, G., Mazzieri, L.: Gradient Einstein solitons. Nonlinear Anal. 132, 66–94 (2016)
De, U.C., Turan, M., Yildiz, A., De, A.: Ricci solitons and gradient Ricci solitons on 3-dimensional normal almost contact metric manifolds. Publ. Math. Debrecen 80, 127–142 (2012)
Cho, J.T.: Almost contact 3-manifolds and Ricci solitons. Int. J. Geom. Methods Mod. Phys. 10(1), 1220022 (2013)
Azami, S., Fasihi-Ramandi, G.: Ricci \(\rho \)-solitons on 3-dimensional \(\eta \)-Einstein Almost Kenmotsu manifolds. Commun. Korean Math. Soc. 35(2), 613–623 (2020)
Siddiqi, M.D., Chaubey, S.K., Khan, M.N.I.: \(f({\cal{R}},{\cal{T}})\)-gravity model with perfect fluid admitting Einstein solitons. Mathematics 10(1), 82
Koufogiorgos, T.: On a class of contact Riemannian 3-manifolds. Results Math. 27(1–2), 51–62 (1995)
Milnor, J.: Curvature of left invariant metrics on Lie groups. Adv. Math. 21, 293–329 (1976)
Sasaki, S.: On differentiable manifolds with certain structures which are closely related to almost contact structure I. Tohoku Math. J. 12(3), 459–476 (1960)
Sasaki, S., Hatakeyama, Y.: On differentiable manifolds with certain structures which are closely related to almost contact structure II. Tohoku Math. J. 13(1), 281–294 (1961)
Blair, D.E.: Riemannian Geometry of Contact and Sympletic Manifolds. Progress in Mathemathics, vol. 203. Birkhäuser, Boston (2010)
Blair, D.E.: Two remarks on contact metric structures. Tohoku Math. J. 29(3), 319–324 (1977)
Calvaruso, G., Perrone, D., Vanhecke, L.: Homogeneity on three-dimensional contact metric manifolds. Isr. J. Math. 114, 301–321 (1999)
Koufogiorgos, T., Markellous, M., Papantoniou, V.J.: The harmonicity of the Reeb vector field on contact metric 3-manifolds. Pac. J. Math. 234(2), 325–344 (2008)
Koufogiorgos, T., Tsichlias, C.: On the existence of a new class of contact metric manifolds. Can. Math. Bull. 43(4), 440–447 (2000)
Chen, X.: Three dimensional contact metric manifolds with Cotton solitons. Hiroshima Math. J. 51, 275–299 (2021)
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Khatri, M., Singh, J.P. Ricci–Bourguignon Soliton on Three-Dimensional Contact Metric Manifolds. Mediterr. J. Math. 21, 70 (2024). https://doi.org/10.1007/s00009-024-02609-0
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DOI: https://doi.org/10.1007/s00009-024-02609-0