Rays and souls in von Mangoldt planes

TL;DR

This paper investigates the geometric properties of rays and souls in von Mangoldt planes, revealing their structure and how they relate to curvature bounds, with applications to manifold geometry.

Contribution

It characterizes the set of souls in rotationally symmetric nonnegative curvature planes and demonstrates how cones in Euclidean space can be smoothed into von Mangoldt planes.

Findings

Set of souls forms a closed ball in such planes

Computed the radius of the soul ball in von Mangoldt planes

Euclidean cones can be smoothed into von Mangoldt planes

Abstract

We study rays in von Mangoldt planes, which has applications to the structure of open complete manifolds with lower radial curvature bounds. We prove that the set of souls of any rotationally symmetric plane of nonnegative curvature is a closed ball, and if the plane is von Mangoldt, we compute the radius of the ball. We show that each cone in the Euclidean 3-space can be smoothed to a von Mangoldt plane.