  We formulate the weak separation condition and the finite type condition for
conformal iterated function systems on Riemannian manifolds with nonnegative
Ricci curvature, and generalize the main theorems by Lau \textit{et al.} in
[Monatsch. Math. 156 (2009), 325-355]. We also obtain a formula for the
Hausdorff dimension of a self-similar set defined by an iterated function
system satisfying the finite type condition, generalizing a corresponding
result by Jin-Yau [Comm. Anal. Geom. 13 (2005), 821--843] and Lau-Ngai [Adv.
Math. 208 (2007), 647-671] on Euclidean spaces. Moreover, we obtain a formula
for the Hausdorff dimension of a graph self-similar set generated by a
graph-directed iterated function system satisfying the graph finite type
condition, extending a result by Ngai \textit{et al.} in [Nonlinearity 23
(2010), 2333--2350].
