Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-24T16:36:33.905Z Has data issue: false hasContentIssue false
  • English
  • Français

QUASI-EINSTEIN CONTACT METRIC MANIFOLDS

Part of: Global differential geometry

Published online by Cambridge University Press:  18 December 2014

AMALENDU GHOSH
AMALENDU GHOSH*
Affiliation:
Department of Mathematics, Chandernagore College, Chandannagar, 712 136, W.B.India e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider quasi-Einstein metrics in the framework of contact metric manifolds and prove some rigidity results. First, we show that any quasi-Einstein Sasakian metric is Einstein. Next, we prove that any complete K-contact manifold with quasi-Einstein metric is compact Einstein and Sasakian. To this end, we extend these results for (κ, μ)-spaces.

MSC classification

Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Secondary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 3 , September 2015 , pp. 569 - 577
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

REFERENCES

1.Apostolov, V., Drǎghichi, T. and Moroianu, A., The odd dimensional Goldberg conjecture, Math. Nachr. 279 (9–10) (2006), 948–952.CrossRefGoogle Scholar
2.Besse, A. L., Einstein manifolds (Springer, Berlin, 1987).CrossRefGoogle Scholar
3.Blair, D. E., Riemannian geometry of contact and symplectic manifolds (Birkhauser, Boston, 2002).CrossRefGoogle Scholar
4.Blair, D. E., Koufogiorgos, T. and Papantoniou, B. J., Contact metric manifolds satisfying a nullity condition, Israel J. Math. 91 (1–3) (1995), 189–214.CrossRefGoogle Scholar
5.Blair, D. E. and Sharma, R., Three dimensional locally symmetric contact metric manifolds, Boll. U.M.I. 4–A (7) (1990), 385–390.Google Scholar
6.Boyer, C. P. and Galicki, K., Einstein manifolds and contact geometry, Proc. Am. Math. Soc. 129 (8) (2001), 2419–2430.CrossRefGoogle Scholar
7.Cao, H. D., Recent progress on Ricci soliton Adv. Lect. Math. 11 (2009), 1–38.Google Scholar
8.Case, J., On the non-existence of quasi-Einstein metrics, Pac. J. Math. 248 (2) (2010), 227–284CrossRefGoogle Scholar
9.Case, J., Shu, Y. and Wei, G., Rigidity of quasi-Einstein metrics, Differ. Geom. Appl. 29 (1) (2010), 93–100.CrossRefGoogle Scholar
10.Cho, J. T. and Sharma, R., Contact geometry and Ricci solitons, Int. J. Geom. Methods Math. Phys. 7 (6) (2010), 951–960.CrossRefGoogle Scholar
11.Ghosh, A., Ricci solitons and contact metric manifolds, Glasgow Math. J. 55 (1) (2013), 123–130.CrossRefGoogle Scholar
12.Ghosh, A., Sharma, R. and Cho, J. T., Contact metric manifolds with η-parallel torsion tensor, Ann. Glob. Anal. Geom. 34 (3) (2008), 287–299.CrossRefGoogle Scholar
13.Hamilton, R. S., The Ricci flow: an introduction in: mathematics and general relativity (Santa Cruz, CA, 1986), Contemporary Mathematics, vol. 71 (American Mathematical Society, Providence RI, 1988), 237–262.Google Scholar
14.Kim, D.-S. and Kim, Y. H., Compact Einstein warped product spaces with non-positive scalar curvature, Proc. Am. Math. Soc. 131 (8) (2003), 2573–2576.CrossRefGoogle Scholar
15.Perelman, G., The entropy formula for the Ricci flow and its geometric applications, Preprint, http:arXiv.org_abs_math.DG/02111159.Google Scholar
16.Petersen, P. and Wylie, W., Rigidity of gradient Ricci solitons, Pacific J. Math. 241 (2) (2009), 329–345.CrossRefGoogle Scholar
17.Qian, Z., Estimates for weighted volumes and applications, Q. J. Math. Oxford Ser. 48 (2) (1997), 235–242.CrossRefGoogle Scholar
18.Sharma, R., Certain results on K-contact and (k,μ)-contact manifolds, J. Geom. 89 (1–2) (2008), 138–147.CrossRefGoogle Scholar
19.Tanno, S., The topology of contact Riemannian manifolds, Illinois J. Math. 12 (4) (1968), 700–717.CrossRefGoogle Scholar
20.Yano, K., Integral formulas in Riemannian geometry (Marcel Dekker, New York, 1970).Google Scholar