  In this paper we study Higgs and co-Higgs $G$-bundles on compact K\"ahler
manifolds $X$. Our main results are: (1) If $X$ is Calabi-Yau, and
$(E,\,\theta)$ is a semistable Higgs or co-Higgs $G$-bundle on $X$, then the
principal $G$-bundle $E$ is semistable. In particular, there is a deformation
retract of ${\mathcal M}_H(G)$ onto $\mathcal M(G)$, where $\mathcal M(G)$ is
the moduli space of semistable principal $G$-bundles with vanishing rational
Chern classes on $X$, and analogously, ${\mathcal M}_H(G)$ is the moduli space
of semistable principal Higgs $G$-bundles with vanishing rational Chern
classes. (2) Calabi-Yau manifolds are characterized as those compact K\"ahler
manifolds whose tangent bundle is semistable for every K\"ahler class, and have
the following property: if $(E,\,\theta)$ is a semistable Higgs or co-Higgs
vector bundle, then $E$ is semistable.
