On Scalar Curvature of r-Almost Newton-Ricci Solitons

The aim of this work is to study the scalar curvature of an r-almost-Newton-Ricci soliton immersed in a Riemannian manifold introduced by Cunha et al. in (J Math Anal Appl 464(I), 546-556, 2018). We study the compact case and obtain an integral formula and some triviality results. Also, we obtain a type Schur inequality. Finally we approach the complete case and obtain constant scalar curvature by assuming some integral conditions on the potential function.