  We present a method to obtain a scalar potential at tree level from a pure
gauge theory on nilmanifolds, a class of negatively-curved compact spaces, and
discuss the spontaneous symmetry breaking mechanism induced in the residual
Minkowski space after compactification at low energy. We show that the scalar
potential is completely determined by the gauge symmetries and the geometry of
the compact manifold. In order to allow for simple analytic calculations we
consider three extra space dimensions as the minimal example of a nilmanifold,
therefore considering a pure Yang-Mills theory in seven dimensions.
