  A bounded domain $\Omega$ in a Riemannian manifold $M$ is said to have the
Pompeiu property if the only continuous function which integrates to zero on
$\Omega$ and on all its congruent images is the zero function. In some
respects, the Pompeiu property can be viewed as an overdetermined problem,
given its relation with the Schiffer problem. It is well-known that every
Euclidean ball fails the Pompeiu property while spherical balls have the
property for almost all radii (Ungar's Freak theorem). In the present paper we
discuss the Pompeiu property when $M$ is compact and admits an isoparametric
foliation. In particular, we identify precise conditions on the spectrum of the
Laplacian on $M$ under which the level domains of an isoparametric function
fail the Pompeiu property. Specific calculations are carried out when the
ambient manifold is the round sphere, and some consequences are derived.
Moreover, a detailed discussion of Ungar's Freak theorem and its
generalizations is also carried out.
