  We study rigidity properties of actions of a torsion-free lattice of
$\operatorname{PSU}(1,1)$ on the circle $S^1$. We follow the approaches of
Frankel and Thurston proposed in preprints via foliated harmonic measures on
the suspension bundles. Our main results are a curvature estimate and a
Gauss--Bonnet formula for the $S^1$ connection obtained by taking the average
of the flat connection with respect to a harmonic measure. As consequences, we
give a precise description of the harmonic measure on suspension foliations
with maximal Euler number and an alternative proof of rigidity theorems of
Matsumoto and Burger--Iozzi--Wienhard.
