# Solitons and geometrical structures in a perfect fluid spacetime

## TL;DR

This paper explores the geometric properties of perfect fluid spacetimes through curvature tensors and solitons, deriving conditions for different types of Ricci solitons and a Poisson equation in the gradient case.

## Contribution

It introduces new conditions for $ ext{η}$-Ricci and $ ext{η}$-Einstein solitons in perfect fluid spacetimes and derives a Poisson equation for gradient solitons.

## Key findings

- Conditions for Ricci solitons to be steady, expanding, or shrinking.
- Derivation of a Poisson equation in the gradient potential case.
- Analysis of geometrical structures via curvature tensors.

## Abstract

Geometrical aspects of a perfect fluid spacetime are described in terms of different curvature tensors and $\eta$-Ricci and $\eta$-Einstein solitons in a perfect fluid spacetime are determined. Conditions for the Ricci soliton to be steady, expanding or shrinking are also given. In a particular case when the potential vector field $\xi$ of the soliton is of gradient type, $\xi:=grad(f)$, we derive a Poisson equation from the soliton equation.