Vol. 288, No. 2, 2017

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ISSN: 0030-8730
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 n+1, in the de Sitter space S1n+1, or in the anti-de Sitter space 1n+1. 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 n, when the ambient space is n+1 or 1n+1, or to Sn, when the ambient space is S1n+1.

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