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.
- Cut locus
- Focal locus
- Open triangle
- Radial curvature
- Riemannian manifold with boundary
- Toponogov's comparison theorem
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