Harmonic aspects in an $\eta$-Ricci soliton

TL;DR

This paper explores the properties of $ ext{eta}$-Ricci solitons with harmonic or Schr"odinger-Ricci harmonic forms, providing conditions for solutions and applying results to perfect fluid spacetimes and topological implications.

Contribution

It characterizes $ ext{eta}$-Ricci solitons with harmonic forms and establishes conditions for solutions to the Schr"odinger-Ricci equation, linking geometric and topological properties.

Findings

Conditions for $ ext{eta}$ to be a Schr"odinger-Ricci harmonic form.

Relations between harmonic, Schr"odinger-Ricci harmonic forms, and $ ext{eta}$-Ricci solitons.

Topological properties of manifolds with these structures.

Abstract

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.