# Rigid properties of generalized $\tau$-quasi Ricci-harmonic metrics

## TL;DR

This paper investigates the properties and rigidity of compact generalized $ au$-quasi Ricci-harmonic metrics, providing conditions for harmonic-Einstein status and establishing gap theorems with specific criteria.

## Contribution

It offers new characterization and rigidity results for generalized $ au$-quasi Ricci-harmonic metrics, including conditions for harmonic-Einstein metrics and gap theorems.

## Key findings

- Conditions under which these metrics are harmonic-Einstein
- Rigidity results for compact $( au,
ho)$-quasi Ricci-harmonic metrics
- Necessary and sufficient conditions for metrics to be harmonic-Einstein

## Abstract

In this paper, we study compact generalized $\tau$-quasi Ricci-harmonic metrics. In the first part, we explore conditions under which generalized $\tau$-quasi Ricci-harmonic metrics are harmonic-Einstein and give some characterization results for it. In the second part, we obtain some rigidity results for compact $(\tau, \rho)$-quasi Ricci-harmonic metrics which are special case of generalized $\tau$-quasi Ricci-harmonic metrics. In the third part, we shall give two gap theorems for compact $\tau$-quasi Ricci-harmonic metrics by showing some necessary and sufficient conditions for the metrics to be harmonic-Einstein.