Applications of Toponogov's comparison theorems for open triangles

Details Applications of Toponogov's comparison theorems for open triangles Applications of Toponogov's comparison theorems for open triangles

2011-02-21

arxiv.org/abs/1102.4156v1

Mathematics

Differential Geometry

This article is a version split from our article "Toponogov comparison theorem for open triangles" (arXiv:0905.3236) to submit

Scientific paper

Recently we generalized Toponogov's comparison theorem to a complete Riemannian manifold with smooth convex boundary, where a geodesic triangle was replaced by an open (geodesic) triangle standing on the boundary of the manifold, and a model surface was replaced by the universal covering surface of a cylinder of revolution with totally geodesic boundary. The aim of this article is to prove splitting theorems of two types as an application. Moreover, we establish a weaker version of our Toponogov comparison theorem for open triangles, because the weaker version is quite enough to prove one of the splitting theorems.

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