LAUSR

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 this case. In the second part, we obtain some rigidity results for compact $$(\tau , \rho )$$ -quasi Ricci-harmonic metrics which are a special case of generalized $$\tau $$ -quasi Ricci-harmonic metrics. In the third part, we 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.