Sólitons de Ricci com estrutura de Produto Deformado

Abstract

In this work we show that either expanding or steady gradient Ricci soliton warped product, whose warping function reaches both maximum and minimum, must be a Riemannian product. Firstly, we present a necessary and sufficient condition for cons-tructing a gradient Ricci soliton warped product. As an application, we give a new class of expanding gradient Ricci soliton warped products having as fiber an Einstein manifold with non-positive scalar curvature. Secondly, we discuss some restrictions to this latter construction, and especially in the case when the base of the warped product is compact. Thirdly, we introduce the modified Ricci solitons as a new class of Einstein type metrics that contains both Ricci solitons and m-quasi-Einstein metrics. This class is closely related to the construction of the Ricci solitons that are realised as warped products. On the other hand, a modified Ricci soliton appears as part of a self-similar solution of the modified Harmonic-Ricci flow which results in a new characterisation of m-quasi-Einstein metrics. Finally, we study a modified almost Ricci soliton. In the spirit of Lichnerowicz and Obata theorems, we prove that in the class of compact Riemannian manifolds with constant scalar curvature the standard sphere with a structure of gradi-ent modified almost Ricci soliton is rigid under some specific geometric condition. An existence condition for constructing an almost Ricci soliton warped product as well as an example of expanding non-gradient Ricci soliton warped product are presented.