# A potential generalization of some canonical Riemannian metrics

## TL;DR

This paper explores new classes of Riemannian manifolds with potential functions, generalizing classical metrics like Einstein and Yamabe, providing classifications, rigidity results, and examples, including nongradient cases.

## Contribution

It introduces a unified framework for various canonical Riemannian metrics with potential functions, extending existing theories and offering new classifications and examples.

## Key findings

- Complete classification of the most rigid cases.
- Rigidity and obstruction results for other cases.
- Descriptions of nongradient potential function constructions.

## Abstract

The aim of this paper is to study new classes of Riemannian manifolds endowed with a smooth potential function, including in a general framework classical canonical structures such as Einstein, harmonic curvature and Yamabe metrics, and, above all, gradient Ricci solitons. For the most rigid cases we give a complete classification, while for the others we provide rigidity and obstruction results, characterizations and nontrivial examples. In the final part of the paper we also describe the "nongradient" version of this construction.