Illinois Journal of Mathematics

Rigidity of gradient almost Ricci solitons

A. Barros, R. Batista, and E. Ribeiro Jr.

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In this paper, we show that either, a Euclidean space $\mathbb{R}^{n}$, or a standard sphere $\mathbb{S}^{n}$, is the unique manifold with nonnegative scalar curvature which carries a structure of a gradient almost Ricci soliton, provided this gradient is a non trivial conformal vector field. Moreover, in the spherical case the field is given by the first eigenfunction of the Laplacian. Finally, we shall show that a compact locally conformally flat almost Ricci soliton is isometric to Euclidean sphere $\mathbb{S}^{n}$ provided an integral condition holds.

Article information

Illinois J. Math., Volume 56, Number 4 (2012), 1267-1279.

First available in Project Euclid: 6 May 2014

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Zentralblatt MATH identifier

Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20] 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]
Secondary: 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]


Barros, A.; Batista, R.; Ribeiro Jr., E. Rigidity of gradient almost Ricci solitons. Illinois J. Math. 56 (2012), no. 4, 1267--1279. doi:10.1215/ijm/1399395831.

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