Remarks on almost $\eta$-Ricci solitons in $(\varepsilon)$-para Sasakian manifolds

Abstract

We consider almost $\eta$-Ricci solitons in $(\varepsilon)$-para Sasakian manifolds satisfying certain curvature conditions. In the gradient case we give an estimation of the Ricci curvature tensor's norm and express the scalar curvature of the manifold in terms of the functions that define the soliton. We also prove that if the Ricci operator is Codazzi, then the gradient $\eta$-Ricci soliton is expanding if $M$ is spacelike or shrinking if $M$ is timelike.