  Dedicated to Professor Gromoll: The aim of our article is to generalize the
Toponogov comparison theorem to a complete Riemannian manifold with smooth
convex boundary. A geodesic triangle will be replaced by an open (geodesic)
triangle standing on the boundary of the manifold, and a model surface will be
replaced by the universal covering surface of a cylinder of revolution with
totally geodesic boundary. Applications of our theorem are found in our article
"Applications of Toponogov's comparison theorems for open triangles"
(arXiv:1102.4156).
