Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-22T22:32:07.474Z Has data issue: false hasContentIssue false
  • English
  • Français

Ricci Solitons on Almost Co-Kähler Manifolds

Part of: Symplectic geometry, contact geometry Global differential geometry

Published online by Cambridge University Press:  07 December 2018

Yaning Wang
Yaning Wang*
Affiliation:
School of Mathematics and Information Sciences, Henan Normal University, Xinxiang 453007, Henan, P. R. China Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

almost co-Kähler manifoldRicci soliton𝜂-EinsteinLie group

MSC classification

Primary: 53D15: Almost contact and almost symplectic manifolds
Secondary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 4 , December 2019 , pp. 912 - 922
Copyright
© Canadian Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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

References

Blair, D. E., The theory of quasi-Sasakian structures . J. Differential Geometry 1(1967), 331–345.Google Scholar
Blair, D. E., Riemannian geometry of contact and symplectic manifolds . Progress in Mathematics, 203, Birkhäuser, Boston, 2010. https://doi.org/10.1007/978-0-8176-4959-3 Google Scholar
Cappelletti-Montano, B., Nicola, A. D., and Yudin, I., A survey on cosymplectic geometry . Rev. Math. Phys. 25(2013), 1343002, 55 pp. https://doi.org/10.1142/S0129055X13430022 Google Scholar
Conti, D. and Fernández, M., Einstein almost cokähler manifolds . Math. Nachr. 289(2016), 1396–1407. https://doi.org/10.1002/mana.201400412 Google Scholar
Cho, J. T., Almost contact 3-manifolds and Ricci solitons . Int. J. Geom. Methods Mod. Phys. 10(2013), 1220022, 7 pp. https://doi.org/10.1142/S0219887812200228 Google Scholar
Hamilton, R. S., Three-manifolds with positive Ricci curvature . J. Differ. Geom. 17(1982), 255–306. https://doi.org/10.4310/jdg/1214436922 Google Scholar
Hamilton, R. S., The Ricci flow on surfaces . Contemp. Math., 71, American Mathematicl Society, Providence, RI, 1988. https://doi.org/10.1090/conm/071/954419 Google Scholar
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), 627–636. https://doi.org/10.11568/kjm.2016.24.4.627 Google Scholar
Li, H., Topology of co-symplectic/co-Kähler manifolds . Asian J. Math. 12(2008), 527–544. https://doi.org/10.4310/AJM.2008.v12.n4.a7 Google Scholar
Montano, B. C. and Pastore, A. M., Einstein-like conditions and cosymplectic geometry . J. Adv. Math. Stud. 3(2010), 27–40.Google Scholar
Olszak, Z., On almost cosymplectic manifolds . Kodai Math. J. 4(1981), 239–250.Google Scholar
Olszak, Z., Locally conformal almost cosymplectic manifolds . Colloq. Math. 57(1989), 73–87. https://doi.org/10.4064/cm-57-1-73-87 Google Scholar
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), 151–161. https://doi.org/10.21099/tkbjm/1496162985 Google Scholar
Perelman, G., The entropy formula for the Ricci flow and its geometric applications. 2012. arxiv:math/0211159 Google Scholar
Perrone, D., Minimal Reeb vector fields on almost cosymplectic manifolds . Kodai Math. J. 36(2013), 258–274. https://doi.org/10.2996/kmj/1372337517 Google Scholar
Wang, W., A class of three dimensional almost co-Kähler manifolds . Palest. J. Math. 6(2017), 111–118.Google Scholar
Wang, W. and Liu, X., Three-dimensional almost co-Kähler manifolds with harmonic Reeb vector field . Rev. Un. Mat. Argentina 58(2017), 307–317.Google Scholar
Wang, Y., A generalization of the Goldberg conjecture for co-Kähler manifolds . Mediterr. J. Math. 13(2016), 2679–2690. https://doi.org/10.1007/s00009-015-0646-8 Google Scholar
Wang, Y., Ricci solitons on 3-dimensional cosymplectic manifolds . Math. Slovaca 67(2017), 979–984. https://doi.org/10.1515/ms-2017-0026 Google Scholar
Yano, K., Integral formulas in Riemannian geometry . Pure and Applied Mathematics, No. 1, Marcel Dekker, New York, 1970.Google Scholar