  This paper considers the relationship between geometry, symmetry and
fundamental interactions -- gravity and those mediated by gauge fields. We
explore product spacetimes which a) have the necessary symmetries for gauge
interactions and four-dimensional gravity and b) reduce to an N-dimensional
isotropic universe in their flat space limit. The key technique is looking at
orbits of the operator form of symmetric rank-two tensors under changes of
coordinate system. Orbits containing diagonal matrices are seen to correspond
to product manifolds. The GL(N, R) symmetry of the decompactified universe acts
non-linearly on such a product spacetime.
  We explore the resulting Kaluza-Klein theories, in which the internal
symmetries act indirectly on space of the extra dimensions, and give two
examples: a six-dimensional model in which the gauge symmetry is U (1) and a
seven-dimensional model in which it is SU (2). We identify constraints that can
be placed on any rank-two symmetric tensor to obtain such spacetimes:
relationships between polynomial invariants. The multiplicities of its
eigenvalues determine the dimensionalities of the factor spaces and hence the
gauge symmetries. If the tensor in question is the Ricci tensor, other than
two-dimensional factor spaces all the factor spaces are Einstein manifolds.
This situation represents the classical vacuum of the Kaluza-Klein theory
