## Illinois Journal of Mathematics

### Rigidity of gradient almost Ricci solitons

#### Abstract

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

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

Dates
First available in Project Euclid: 6 May 2014

https://projecteuclid.org/euclid.ijm/1399395831

Digital Object Identifier
doi:10.1215/ijm/1399395831

Mathematical Reviews number (MathSciNet)
MR3231482

Zentralblatt MATH identifier
1290.53053

#### Citation

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. https://projecteuclid.org/euclid.ijm/1399395831

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