We prove a version of Topogonov's triangle comparison theorem with
surfaces of revolution as model spaces.

Given a model surface and a
Riemannian manifold with a fixed base point, we give necessary and
sufficient conditions under which every geodesic
triangle in the manifold with a vertex at the base point has a
corresponding Alexandrov triangle in the model.

Under these conditions
we also prove a version of the Maximal Radius Theorem and present two
topological applications.

This is joint work with
Yutaka Ikeda.