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

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.