# A new approach to gradient Ricci solitons and generalizations

## TL;DR

This paper explores inequalities relating curvature and scalar fields in gradient Ricci solitons and introduces generalizations to manifolds with linear connections beyond metric types.

## Contribution

It presents new inequalities in Ricci soliton geometry and extends the concept to manifolds with non-metric linear connections.

## Key findings

- Derived inequalities linking curvature and scalar functions in Ricci solitons
- Established relationships between curvature and scalar field behavior
- Proposed generalizations to non-metric linear connection manifolds

## Abstract

This short note concerns with two inequalities in the geometry of gradient Ricci solitons $(g, f, \lambda )$ on a smooth manifold $M$. These inequalities provide some relationships between the curvature of the Riemannian metric $g$ and the behavior of the scalar field $f$ through two second order equations satisfied by the scalar $\lambda $. We propose several generalizations of Ricci solitons to the setting of manifolds endowed with linear connections, not necessary of metric type.