We formulate the weak separation condition and finite type for conformal iterated function systems on Riemannian manifolds with nonnegative Ricci curvature, generalize main theorems by Lau et al. (Monatsch Math 156:325–355, 2009). also obtain a formula Hausdorff dimension of self-similar set defined an system satisfying condition, generalizing corresponding result Jin Yau (Commun Anal Geom 13:8...