Existence of quasi-isometric mappings and Royden compactifications

It is shown that Royden p-compactifications for 1 < p < d of two orientable and countable Riemannian manifolds of the same dimension d greater than or equal to 2 are homeomorphic if and only if there exists a homeomorphism between these two Riemannian manifolds which is quasi-isometric outside some of their compact subsets.