A variational principle for Kaluza-Klein type theories

For any positive integer $n$ and any Lie group $\mathfrak{G}$, given a definite symmetric bilinear form on ${\mathbb{R}}^n$ and an $\text{Ad}$-invariant scalar product on the Lie algebra of $\mathfrak{G}$, we construct a variational problem on fields defined on an arbitrary oriented $(n+\text{dim}\mathfrak{G})$-dimensional manifold $\mathcal{Y}$. We show that, if $\mathfrak{G}$ is compact and simply connected, any global solution of the Euler–Lagrange equations leads, through a spontaneous symmetry breaking, to identify $\mathcal{Y}$ with the total space of a principal bundle over an $n$-dimensional manifold $\mathcal{X}$. Moreover $\mathcal{X}$ is then endowed with a (pseudo-)Riemannian metric and a connection which are solutions of the Einstein–Yang–Mills system of equations with a cosmological constant.