Variations on Gromov's open-dense orbit theorem

We investigate several situations where the local homogeneity of a geometric structure on a dense open subset of a manifold implies the local homogeneity everywhere. This results in a strengthening of the conclusions in Gromov’s open-dense orbit theorem. In particular, we show that any smooth closed 3-dimensional Lorentz manifold with a topologically transitive isometric action must be locally homogeneous.