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

DOI: 10.2140/pjm.2017.288.289

Abstract

Our purpose is to study the geometry of gradient almost Ricci solitons isometrically immersed either in the hyperbolic space ${ℍ}^{n+1}$, in the de Sitter space ${\mathbb{S}}_1^{n+1}$, or in the anti-de Sitter space ${ℍ}_1^{n+1}$. 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 ${ℍ}^n$, when the ambient space is ${ℍ}^{n+1}$ or ${ℍ}_1^{n+1}$, or to ${\mathbb{S}}^n$, when the ambient space is ${\mathbb{S}}_1^{n+1}$.

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