Inscriptions in non-Euclidean Geometries

2025/07/10 by Ali Naseri Sadr, Sadr, Ali Naseri
Mathematics · #Geometric Analysis and Curvature Flows #Algebraic and Geometric Analysis #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2507.07945

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.

Citations

• Periodic Inscription of Isosceles Trapezoids
• Cyclic quadrilaterals and smooth Jordan curves
• The Rectangular Peg Problem
• Lagrangian embeddings of the Klein bottle and combinatorial properties of mapping class groups
• A remark on Kähler forms on symmetric products of Riemann surfaces

Related

• Inscribed triangles of Jordan curves in ℝⁿ
• Inscribed Squares and Relation Avoiding Paths
• Constant Angle Ruled Surfaces in Euclidean Spaces
• Square-like quadrilaterals inscribed in embedded space curves
• Non-orientable slice surfaces and inscribed rectangles