Harmonic aspects in an $\eta$-Ricci soliton

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 Schrödinger-Ricci harmonic form. We also provide necessary and sufficient conditions for $\eta$ to be a solution of the Schrödinger-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- Weitzenböck techniques, we formulate some more conclusions for gradient solitons and deduce topological properties of the manifold and its universal covering.

17 pages

References in corpus (2)

• $\eta$-Ricci solitons in $(\epsilon )$-almost paracontact metric manifolds
• Remarks on almost $\eta$-Ricci solitons in $(\epsilon )$-para Sasakian manifolds