$r$-Almost Newton-Ricci solitons immersed in a Lorentzian manifold: examples, nonexistence and rigidity

Abstract

We establish the concept of $r$-almost Newton-Ricci soliton immersed into a Lorentzian manifold, which extends in a natural way the almost Ricci solitons introduced by Pigola, Rigoli, Rimoldi and Setti in [17]. In this setting, under suitable hypothesis on the potential and soliton functions, we obtain nonexistence and rigidity results. Some interesting examples of these new geometric objects are also given.