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Ricci Solitons on Almost Co-Kähler Manifolds

  • Yaning Wang (a1)


In this paper, we prove that if an almost co-Kähler manifold of dimension greater than three satisfying $\unicode[STIX]{x1D702}$ -Einstein condition with constant coefficients is a Ricci soliton with potential vector field being of constant length, then either the manifold is Einstein or the Reeb vector field is parallel. Let $M$ be a non-co-Kähler almost co-Kähler 3-manifold such that the Reeb vector field $\unicode[STIX]{x1D709}$ is an eigenvector field of the Ricci operator. If $M$ is a Ricci soliton with transversal potential vector field, then it is locally isometric to Lie group $E(1,1)$ of rigid motions of the Minkowski 2-space.



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This work was supported by Youth Science Foundation of Henan Normal University (No. 2014QK01).



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[1] Blair, D. E., The theory of quasi-Sasakian structures . J. Differential Geometry 1(1967), 331345.
[2] Blair, D. E., Riemannian geometry of contact and symplectic manifolds . Progress in Mathematics, 203, Birkhäuser, Boston, 2010.
[3] Cappelletti-Montano, B., Nicola, A. D., and Yudin, I., A survey on cosymplectic geometry . Rev. Math. Phys. 25(2013), 1343002, 55 pp.
[4] Conti, D. and Fernández, M., Einstein almost cokähler manifolds . Math. Nachr. 289(2016), 13961407.
[5] Cho, J. T., Almost contact 3-manifolds and Ricci solitons . Int. J. Geom. Methods Mod. Phys. 10(2013), 1220022, 7 pp.
[6] Hamilton, R. S., Three-manifolds with positive Ricci curvature . J. Differ. Geom. 17(1982), 255306.
[7] Hamilton, R. S., The Ricci flow on surfaces . Contemp. Math., 71, American Mathematicl Society, Providence, RI, 1988.
[8] Lee, S. D., Byung, H. K., and Choi, J. H., On a classification of warped product spaces with gradient Ricci solitons . Korean J. Math. 24(2016), 627636.
[9] Li, H., Topology of co-symplectic/co-Kähler manifolds . Asian J. Math. 12(2008), 527544.
[10] Montano, B. C. and Pastore, A. M., Einstein-like conditions and cosymplectic geometry . J. Adv. Math. Stud. 3(2010), 2740.
[11] Olszak, Z., On almost cosymplectic manifolds . Kodai Math. J. 4(1981), 239250.
[12] Olszak, Z., Locally conformal almost cosymplectic manifolds . Colloq. Math. 57(1989), 7387.
[13] Oguro, T. and Sekigawa, K., Almost Kähler structures on the Riemannian product of a 3-dimensional hyperbolic space and a real line . Tsukuba J. Math. 20(1996), 151161.
[14] Perelman, G., The entropy formula for the Ricci flow and its geometric applications. 2012. arxiv:math/0211159
[15] Perrone, D., Minimal Reeb vector fields on almost cosymplectic manifolds . Kodai Math. J. 36(2013), 258274.
[16] Wang, W., A class of three dimensional almost co-Kähler manifolds . Palest. J. Math. 6(2017), 111118.
[17] Wang, W. and Liu, X., Three-dimensional almost co-Kähler manifolds with harmonic Reeb vector field . Rev. Un. Mat. Argentina 58(2017), 307317.
[18] Wang, Y., A generalization of the Goldberg conjecture for co-Kähler manifolds . Mediterr. J. Math. 13(2016), 26792690.
[19] Wang, Y., Ricci solitons on 3-dimensional cosymplectic manifolds . Math. Slovaca 67(2017), 979984.
[20] Yano, K., Integral formulas in Riemannian geometry . Pure and Applied Mathematics, No. 1, Marcel Dekker, New York, 1970.
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Ricci Solitons on Almost Co-Kähler Manifolds

  • Yaning Wang (a1)


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