Abstract
We show that if a compact K-contact metric is a gradient Ricci almost soliton, then it is isometric to a unit sphere S 2n+1. Next, we prove that if the metric of a non-Sasakian (κ, μ)-contact metric is a gradient Ricci almost soliton, then in dimension 3 it is flat and in higher dimensions it is locally isometric to E n+1 × S n (4). Finally, a couple of results on contact metric manifolds whose metric is a Ricci almost soliton and the potential vector field is point wise collinear with the Reeb vector field of the contact metric structure were obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aquino C., Barros A., Ribeiro E. Jr.: Some applications of the Hodge–de Rahm decomposition to Ricci solitons. Results Math. 60, 245–254 (2011)
Barros A., Ribeiro E. Jr.: Some characterizations for compact almost Ricci solitons. Proc. Am. Math. Soc. 140(3), 213–223 (2012)
Barros, A., Batista, R., Ribeiro, E., Jr.: Compact almost Ricci solitons with constant scalar curvature are gradient, Preprint, arXiv:1209.2720v1 [math DG] (2012)
Blair D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Birkhauser, Boston (2002)
Blair D.E., Koufogiorgos T., Papantoniou B.J.: Contact metric manifolds satisfying a nullity condition. Israel J. Math. 91, 189–214 (1995)
Boyer C.P., Galicki K.: Einstein manifolds and contact geometry. Proc. Am. Math. Soc. 129, 2419–2430 (2001)
Boyer C.P., Galicki K.: Sasakian Geometry. Oxford University Press, Oxford (2008)
Cho J.T., Sharma R.: Contact geometry and Ricci solitons. Int. J. Geom. Methods Math. Phy. 7(6), 951–960 (2010)
Ghosh, A.: Ricci solitons and contact metric manifolds. Glasgow Math. J. (2012) (published on line)
Ghosh A., Sharma R., Cho J.T.: Contact metric manifolds with η-parallel torsion tensor. Ann. Glob Anal. Geom. 34, 287–299 (2008)
Hamilton, R.S.: The Ricci flow: an introduction. In: Mathematics and General Relativity (Santa Cruz, CA, 1986), pp. 237–262. Contemp. Math., vol. 71. American Math.Soc., New York (1988)
Hasegawa, I., Seino, M.: Some remarks on Sasakian geometry-applications of Myer’s theorem and the canonical affine connection. J. Hokkaido Univ. Educ. 32(section IIA), 1–7 (1981)
Obata M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Japan 14, 333–340 (1962)
Olszak Z.: On contact metric manifolds. Tohoku Math. J. 34, 247–253 (1979)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications, Preprint, http://arXiv.orgabsmath.DG/02111159
Perrone D.: Homogeneous contact Riemannian three-manifols. Illinois J. Math. 42, 243–256 (1998)
Perrone D.: Contact metric manifolds whose characteristic vector field is a harmonic vector field. Differ. Geom. Appl. 20, 367–378 (2004)
Pigola S., Rigoli M., Rimoldi M., Setti A.: Ricci almost solitons. Ann. Scuola. Norm. Sup. Pisa. CL Sc. (5) X, 757–799 (2011)
Sharma R.: Certain results on K-contact and (k,μ)-contact manifolds. J. Geom. 89, 138–147 (2008)
Tanno S.: The topology of contact Riemannian manifolds. Illinois J. Math. 12, 700–717 (1968)
Yano K.: Integral Formulas in Riemannian Geometry. Marcel Dekker, New York (1970)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of my father.
Rights and permissions
About this article
Cite this article
Ghosh, A. Certain Contact Metrics as Ricci Almost Solitons. Results. Math. 65, 81–94 (2014). https://doi.org/10.1007/s00025-013-0331-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-013-0331-9