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In this paper, we study $$\Xi $$ -curvature and H-curvature of a special class of Finsler metrics called general $$(\alpha ,\beta )$$ -metrics. We prove that every general $$(\alpha ,\beta )$$ -metric of almost vanishing H-curvature is of almost vanishing $$ \Xi $$ -curvature under certain conditions. Moreover, we study such Finsler metric with vanishing $$\Xi $$ -curvature and its interaction to the flag curvature.

Projective Ricci curvature is a projective invariant quantity in Finsler geometry which is introduced by Z. Shen. In this paper, we study special projective Ricci curvature of C-reducible Finsler metrics. The necessary and sufficient conditions of these metrics, which cause these metrics to be weak or isotropic projective Ricci curvature, are found and it is proved that C-reducible Douglas metric of isotropic PRic-curvature must be PRic flat. The same theorem for C-reducible metrics of scalar flag curvature is also investigated.