Applications of Toponogov's comparison theorems for open triangles

TL;DR

This paper extends Toponogov's comparison theorem to manifolds with boundary using open triangles and applies it to prove splitting theorems, advancing geometric analysis in Riemannian geometry.

Contribution

It introduces a generalized comparison theorem for open triangles on manifolds with boundary and demonstrates its application in proving splitting theorems.

Findings

Established a generalized Toponogov comparison theorem for open triangles.

Proved splitting theorems using the new comparison framework.

Developed a weaker version of the comparison theorem sufficient for certain applications.

Abstract

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.