  We investigate the structure of the configuration space of gauge theories and
its description in terms of the set of absolute minima of certain Morse
functions on the gauge orbits. The set of absolute minima that is obtained when
the background connection is a pure gauge is shown to be isomorphic to the
orbit space of the pointed gauge group. We also show that the stratum of
irreducible orbits is geodesically convex, i.e. there are no geometrical
obstructions to the classical motion within the main stratum. An explicit
description of the singularities of the configuration space of SU(2) theories
on a topologically simple space-time and on the lattice is obtained; in the
continuum case we find that the singularities are conical and that the singular
stratum is isomorphic to a Z_2 orbifold of the configuration space of
electrodynamics.
