  We extend Michel's theorem on the geometry of symmetry breaking [L. Michel,
{\it Comptes Rendus Acad. Sci. Paris} {\bf 272-A} (1971), 433-436] to the case
of pure gauge theories, i.e. of gauge-invariant functionals defined on the
space ${\cal C}$ of connections of a principal fiber bundle. Our proof follows
closely the original one by Michel, using several known results on the geometry
of ${\cal C}$. The result (and proof) is also extended to the case of gauge
theories with matter fields.
