The geometry of Euclidean surfaces with conical singularities

TL;DR

This paper investigates the geometric and dynamical properties of Euclidean surfaces with conical singularities, revealing that non-closed geodesics stay arbitrarily close to the conical points and analyzing geodesic space dynamics.

Contribution

It provides new insights into the behavior of geodesics on Euclidean surfaces with conical singularities and explores their dynamical properties.

Findings

Non-closed geodesics have zero distance from conical points.

Dynamical properties of the geodesic space are characterized.

The image of a non-closed geodesic accumulates near conical points.

Abstract

The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the image of a non-closed geodesic has 0 distance from the set of conical points. Dynamical properties for the space of geodesics are also proved.