#### Vol. 288, No. 2, 2017

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Characterizations of immersed gradient almost Ricci solitons

### Cícero P. Aquino, Henrique F. de Lima and José N. V. Gomes

Vol. 288 (2017), No. 2, 289–305
##### Abstract

Our purpose is to study the geometry of gradient almost Ricci solitons isometrically immersed either in the hyperbolic space ${ℍ}^{n+1\phantom{\rule{0.3em}{0ex}}}$, in the de Sitter space ${\mathbb{S}}_{1}^{n+1}\phantom{\rule{0.3em}{0ex}}$, or in the anti-de Sitter space ${ℍ}_{1}^{n+1}\phantom{\rule{0.3em}{0ex}}$. In each one of these ambient spaces we obtain extensions of a classical theorem due to Nomizu and Smith. More precisely, we show that the totally umbilical hypersurfaces are the only immersed hypersurfaces of such ambient spaces which admit a structure of gradient almost Ricci soliton via the tangential component of a certain fixed vector, and whose image of the Gauss mapping is also totally umbilical. Furthermore, in the case that the structure of gradient almost Ricci soliton is nontrivial, we conclude that such a hypersurface must be isometric either to ${ℍ}^{n}\phantom{\rule{0.3em}{0ex}}$, when the ambient space is ${ℍ}^{n+1}$ or ${ℍ}_{1}^{n+1}\phantom{\rule{0.3em}{0ex}}$, or to ${\mathbb{S}}^{n}\phantom{\rule{0.3em}{0ex}}$, when the ambient space is ${\mathbb{S}}_{1}^{n+1}\phantom{\rule{0.3em}{0ex}}$.

##### Keywords
almost Ricci solitons, hyperbolic space, de Sitter space, anti-de Sitter space, mean curvature, Gauss mapping
##### Mathematical Subject Classification 2010
Primary: 53C42
Secondary: 53B30, 53C50, 53Z05, 83C99
##### Milestones
Received: 13 October 2015
Revised: 18 November 2016
Accepted: 19 December 2016
Published: 28 April 2017
##### Authors
 Cícero P. Aquino Departamento de Matemática Universidade Federal do Piauí 64049-550 Teresina Brazil Henrique F. de Lima Departamento de Matemática Universidade Federal de Campina Grande Av. Aprigio Veloso 882 Bloco CX Bodocongo 58429-900 Campina Grande Brazil José N. V. Gomes Departamento de Matemática Universidade Federal do Amazonas 69077-070 Manaus Brazil