# On warped product gradient $\eta$-Ricci solitons

## TL;DR

This paper investigates gradient $ ext{η}$-Ricci solitons on warped product manifolds, deriving key equations using Bochner formula, and explores conditions under which these solitons are uniquely determined or influence scalar curvature bounds.

## Contribution

It provides a new method to construct gradient $ ext{η}$-Ricci solitons on warped products and links soliton properties to scalar curvature in specific geometric settings.

## Key findings

- Derived Laplacian equation for the potential function $f$
- Proved soliton is determined by $f$ in irrotational case
- Established scalar curvature bounds for compact base manifolds

## Abstract

If the potential vector field of an $\eta$-Ricci soliton is of gradient type, using Bochner formula, we derive from the soliton equation a Laplacian equation satisfied by the potential function $f$. In a particular case of irrotational potential vector field we prove that the soliton is completely determined by $f$. We give a way to construct a gradient $\eta$-Ricci soliton on a warped product manifold and show that if the base manifold is oriented, compact and of constant scalar curvature, the soliton on the product manifold gives a lower bound for its scalar curvature.