Evolutes and isoperimetric deficit in two-dimensional spaces of constant curvature

TL;DR

This paper explores the relationship between curvature, isoperimetric deficit, and evolutes in 2D spaces of constant curvature, offering new geometric insights and a Gauss-Bonnet theorem for specific evolutes.

Contribution

It establishes a novel connection between total curvature, isoperimetric deficit, and evolutes in curved spaces, and introduces a Gauss-Bonnet theorem for certain evolutes.

Findings

Relation between curvature and isoperimetric deficit via evolutes

A Gauss-Bonnet theorem for special evolutes in curved spaces

New geometric formulas linking curvature, area, and evolutes

Abstract

We relate the total curvature and the isoperimetric deficit of a curve $\gamma$ in a two-dimensional space of constant curvature with the area enclosed by the evolute of $\gamma$. We provide also a Gauss-Bonnet theorem for a special class of evolutes.