  We extend some of the results obtained for subvarieties of the moduli stack
of canonically polarized manifolds in "Base spaces of non-isotrivial families
of smooth minimal models" (math.AG/0103122) to moduli of polarized minimal
models of Kodaira dimension zero. To this aim we first show that for those
manifolds there are no non isotrivial families of minimal models, which are
birationally isotrivial. In the last chapter, we discuss the rigidity of curves
in the moduli stack of canonically polarized manifolds and of polarized minimal
models of Kodaira dimension zero.
