r-Almost Newton-Ricci solitons immersed into a Riemannian manifold

We establish the new concept of r-almost Newton-Ricci soliton for hypersurfaces immersed in a Riemannian manifold, which involves the r-th Newton and the Ricci tensors and extends in a natural way the notion of immersed almost Ricci solitons introduced by Barros et al. [3]. In this setting, our purpose is to investigate the existence of these geometric objects. After exhibit some examples of r-almost Newton-Ricci solitons, we obtain sufficient conditions to guarantee that they must be totally geodesic under suitable constraints on the potential function and using appropriate maximum principles. Furthermore, a particular study of 1-almost Newton-Ricci solitons immersed in a locally symmetric Einstein manifold is also made.