  If spacetime possesses extra dimensions of size and curvature radii much
larger than the Planck or string scales, the dynamics of these extra dimensions
should be governed by classical general relativity. We argue that in general
relativity, it is highly nontrivial to obtain solutions where the extra
dimensions are static and are dynamically stable to small perturbations. We
also illustrate that intuition on equilibrium and stability built up from
non-gravitational physics can be highly misleading. For all static, homogeneous
solutions satisfying the null energy condition, we show that the Ricci
curvature of space must be nonnegative in all directions. Much of our analysis
focuses on a class of spacetime models where space consists of a product of
homogeneous and isotropic geometries. A dimensional reduction of these models
is performed, and their stability to perturbations that preserve the spatial
symmetries is analyzed. We conclude that the only physically realistic examples
of classically stabilized large extra dimensions are those in which the
extra-dimensional manifold is positively curved.
