Geometric structures on manifolds and holonomy-invariant metrics

Abstract

Let a manifold M have a geometric structure modelled on the pair (G, X) of a Lie group G and a manifold X on which G acts; that is, the universal cover (M) over tilde of M has an immersion dev: (M) over tilde --> X equivariant with respect to the holonomy homomorphism h:pi(1)(M) --> G. If the image of dev intersects an open subset U of X that has a complete h(pi(1)(M))-invariant metric with certain properties, then dev(-1) (U) covers U under dev, and covers an open submanifold of M under the covering map (M) over tilde --> M. This fills the gap in the proofs of the results of Faltings and Goldman on real and complex projective surfaces.