The Poisson equation on compact Ricci solitons and Ricci-harmonic solitons

In this short note, we derive another proof of a theorem due to Chen and Deshmukh (Balk J Geom Appl 19(1):13-21, 2014) on a characterization of compact shrinking trivial Ricci solitons with scalar curvature satisfying a Poisson equation. Moreover, as a new result, we recall that a version of this theorem holds to the case of compact domain manifolds of shrinking Ricci-harmonic solitons.