  We study maximal $m$-rigid objects in the $m$-cluster category $\mathcal
C_H^m$ associated with a finite dimensional hereditary algebra $H$ with $n$
nonisomorphic simple modules. We show that all maximal $m$-rigid objects in
these categories have exactly $n$ nonisomorphic indecomposable summands, and
that any almost complete $m$-rigid object in $\mathcal C_H^m$ has exactly $m+1$
nonisomorphic complements. We also show that the maximal $m$-rigid objects and
the $m$-cluster tilting objects in these categories coincide, and that the
class of finite dimensional algebras associated with maximal $m$-rigid objects
is closed under certain factor algebras.
