  For any subgroup G of O(n), define a "G-manifold" to be an n-dimensional
Riemannian manifold whose holonomy group is contained in G. Then a G-manifold
where G is the Standard Model gauge group is precisely a Calabi-Yau manifold of
10 real dimensions whose tangent spaces split into orthogonal 4- and
6-dimensional subspaces, each preserved by the complex structure and parallel
transport. In particular, the product of Calabi-Yau manifolds of dimensions 4
and 6 gives such a G-manifold. Moreover, any such G-manifold is naturally a
spin manifold, and Dirac spinors on this manifold transform in the
representation of G corresponding to one generation of Standard Model fermions
and their antiparticles.
