A new approach to gradient Ricci solitons and generalizations

Abstract

This short note concerns with two inequalities in the geo\-me\-try of gradient Ricci solitons $(g, f, \lambda )$ on a smooth manifold $M$. These inequalities provide some relationships between the curvature of the Riemannian metric $g$ and the behavior of the scalar field $f$ through two second order equations satisfied by the scalar $\lambda $. We propose several ge\-ne\-ra\-li\-za\-tions of Ricci solitons to the setting of manifolds endowed with linear connections, not necessary of metric type.