  We present variational formulations of gauge theories and
Einstein--Yang-Mills equations in the spirit of Kaluza-Klein theories. For
gaugetheories, only a topological fibration is assumed. For gravitation coupled
with gauge fields, no fibration is assumed: Fields are defined on a
'space-time' $Y$ of dimension $4 + r$ without any structure a priori, where $r$
is the dimension of the structure group. If the latter is compact and simply
connected, classical solutions allow to construct a manifold $X$ of dimension 4
to be the physical space-time, in such a way that $Y$ acquires the structure of
a principal bundle over $X$ and leads to solutions of the Einstein--Yang-Mills
systems. The special case of the Einstein-Maxwell system is also discussed: It
suffices that at least one fiber closes in on a circle to deduce that the
five-dimensional space-time has afiber bundle structure.
