  We characterize $\eta$-Ricci solitons $(g,\xi,\lambda,\mu)$ in some special
cases when the $1$-form $\eta$, which is the $g$-dual of $\xi$, is a harmonic
or a Schr\"{o}dinger-Ricci harmonic form. We also provide necessary and
sufficient conditions for $\eta$ to be a solution of the Schr\"{o}dinger-Ricci
equation and point out the relation between the three notions in our context.
In particular, we apply these results to a perfect fluid spacetime and using
Bochner- Weitzenb\"{o}ck techniques, we formulate some more conclusions for
gradient solitons and deduce topological properties of the manifold and its
universal covering.
