Massless Kaluza-Klein gauge fields and space-time compactification induced by scalars

It is considered the space-time compactification induced by a general nonlinear sigma-model. A very general class of solutions is given by submersions from the extradimensional space onto the space in which the scalar fields take values. The gauge fields associated with the vertical Killing vectors are massless. It is shown that if the extra space is Riemannian, compact, without boundary, the massless gauge fields admit only Abelian symmetries