Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T01:54:20.496Z Has data issue: false hasContentIssue false
  • English
  • Français

Some results on Ricci-Bourguignon solitons and almost solitons

Part of: Global differential geometry

Published online by Cambridge University Press:  20 August 2020

Shubham Dwivedi
Shubham Dwivedi*
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ONN2L 3G1, Canada
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove some results for the solitons of the Ricci–Bourguignon flow, generalizing the corresponding results for Ricci solitons. Taking motivation from Ricci almost solitons, we then introduce the notion of Ricci–Bourguignon almost solitons and prove some results about them that generalize previous results for Ricci almost solitons. We also derive integral formulas for compact gradient Ricci–Bourguignon solitons and compact gradient Ricci–Bourguignon almost solitons. Finally, using the integral formula, we show that a compact gradient Ricci–Bourguignon almost soliton is isometric to a Euclidean sphere if it has constant scalar curvature or its associated vector field is conformal.

Keywords

Ricci–Bourguignon flowsolitonsalmost solitonsself-similar solutions

MSC classification

Secondary: 53C20: Global Riemannian geometry, including pinching 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 3 , September 2021 , pp. 591 - 604
Check for updates
Copyright
© Canadian Mathematical Society 2020

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.)

References

Aquino, C., Barros, A., and Ribeiro, E. Jr., Some applications of the Hodge–de Rham decomposition to Ricci solitons . Results Math. 60(2011), nos. 1–4, 245–254. https://doi.org/10.1007/s00025-011-0166-1 CrossRefGoogle Scholar
Aubin, T., Métriques riemanniennes et courbure . J. Differ. Geom. 4(1970), 383–424.CrossRefGoogle Scholar
Barros, A. and Ribeiro, E. Jr., Some characterizations for compact almost Ricci solitons . Proc. Am. Math. Soc. 140(2012), no. 3, 1033–1040. https://doi.org/10.1090/S0002-9939-2011-11029-3 CrossRefGoogle Scholar
Bourguignon, J. P., Ricci curvature and Einstein metrics . In: Global differential geometry and global analysis (Berlin, 1979), Lecture Notes in Mathematics, 838, Springer, Berlin, Germany, New York, NY, 1981, pp. 42–63.CrossRefGoogle Scholar
Catino, G., Cremaschi, L., Djadli, Z., Mantegazza, C., and Mazzieri, L., The Ricci-Bourguignon flow . Pac. J. Math. 287(2017), no. 2, 337–370. https://doi.org/10.2140/pjm.2017.287.337 CrossRefGoogle Scholar
Catino, G. and Mazzieri, L., Gradient Einstein solitons . Nonlinear Anal. 132(2016), 66–94. https://doi.org/10.1016/j.na.2015.10.021 CrossRefGoogle Scholar
Catino, G., Mazzieri, L., and Mongodi, S., Rigidity of gradient Einstein shrinkers . Commun. Contemp. Math. 17(2015), no. 6, 1550046, 18 pp. https://doi.org/10.1142/S0219199715500467 CrossRefGoogle Scholar
Lichnerowicz, A., Géométrie des groupes de transformations. Travaux et Recherches Mathématiques, III, Dunod, Paris, 1958.Google Scholar
Obata, M., Certain conditions for a Riemannian manifold to be isometric with a sphere . J. Math. Soc. Jpn. 14(1962), 333–340. https://doi.org/10.2969/jmsj/0140333 CrossRefGoogle Scholar
Petersen, P. and Wylie, W., Rigidity of gradient Ricci solitons . Pac. J. Math. 241(2009), no. 2, 329–345. https://doi.org/10.2140/pjm.2009.241.329 CrossRefGoogle Scholar
Pigola, S., Rigoli, M., Rimoldi, M., and Setti, A. G., Ricci almost solitons . Ann. Sc. Norm. Super. Pisa Cl. Sci. 10(2011), no. 4, 757–799.Google Scholar
Tashiro, Y., Complete Riemannian manifolds and some vector fields . Trans. Am. Math. Soc. 117(1965), 251–275. https://doi.org/10.2307/1994206 CrossRefGoogle Scholar
Yano, K., Integral formulas in Riemannian geometry. Pure and Applied Mathematics, 1, Marcel Dekker, Inc., New York, NY, 1970.Google Scholar