Inscriptions in non-Euclidean Geometries

TL;DR

This paper generalizes classical inscription problems from Euclidean to non-Euclidean geometries, specifically hyperbolic and spherical surfaces, using advanced geometric methods to establish new theorems for smooth Jordan curves.

Contribution

It introduces a framework for inscription problems in Riemannian surfaces of constant curvature and proves new theorems in hyperbolic and spherical geometries using symplectic and Riemannian techniques.

Findings

Inscription theorems for smooth Jordan curves in hyperbolic plane

Rectangular inscription theorem on the sphere for non-antipodal curves

Extension of Euclidean inscription problems to non-Euclidean geometries

Abstract

We show how inscription problems in the plane can be generalized to Riemannian surfaces of constant curvature. We then use ideas from symplectic and Riemannian geometry to prove these generalized versions for smooth Jordan curves in the hyperbolic plane, and we prove a rectangular inscription theorem for smooth Jordan curves on the two sphere that do not intersect their antipodal.