  In a previous work, we studied isoparametric functions on Riemannian
manifolds, especially on exotic spheres. One result there says that, in the
family of isoparametric hypersurfaces of a closed Riemannian manifold, there
exist at least one minimal isoparametric hypersurface. In this note, we show
such minimal isoparametric hypersurface is also unique in the family if the
ambient manifold has positive Ricci curvature. Moreover, we give a proof of the
Theorem D claimed by Q.M.Wang (without proof) which asserts that the focal
submanifolds of an isoparametric function on a complete Riemannian manifold are
minimal. Further, we study isoparametric hypersurfaces with constant principal
curvatures in general Riemannian manifolds. It turns out that in this case the
focal submanifolds have the same properties as those in standard sphere, i.e.,
the shape operator with respect to any normal direction has common constant
principal curvatures. Some necessary conditions involving Ricci curvature and
scalar curvature are also derived.
