Some remarks and results on $h$-almost Ricci solitons

In this paper, we will focus our attention on the structure of $h$-almost Ricci solitons. We obtain certain conditions that if $(M,g)$ be a complete $h$-almost Ricci soliton Riemannian manifold then the fundamental group $\pi_{1}(M)$ of M will finite. Also, we prove that a complete shrinking h-almost Ricci soliton $(M,g,X,h,\lambda)$ is compact if and only if $\| X \|$ is bounded on $(M,g)$.