  Geometric sigma models are purely geometric theories of scalar fields coupled
to gravity. Geometrically, these scalars represent the very coordinates of
space-time, and, as such, can be gauged away. A particular theory is built over
a given metric field configuration which becomes the vacuum of the theory.
Kaluza-Klein theories of the kind have been shown to be free of the classical
cosmological constant problem, and to give massless gauge fields after
dimensional reduction. In this paper, the consistency of dimensional reduction,
as well as the stability of the internal excitations, are analyzed. Choosing
the internal space in the form of a group manifold, one meets no
inconsistencies in the dimensional reduction procedure. As an example, the
SO(n) groups are analyzed, with the result that the mass matrix of the internal
excitations necessarily possesses negative modes. In the case of coset spaces,
the consistency of dimensional reduction rules out all but the stable mode,
although the full vacuum stability remains an open problem.
