Isoperimetric inequalities for wave fronts and a generalization of Menzin's conjecture for bicycle monodromy on surfaces of constant curvature

TL;DR

This paper extends isoperimetric inequalities to spherical and hyperbolic wave fronts, explores bicycle curves on curved surfaces, and proves hyperbolic and spherical versions of Menzin's conjecture, advancing geometric understanding in these contexts.

Contribution

It generalizes isoperimetric inequalities for wave fronts on curved surfaces and resolves Menzin's conjecture in hyperbolic and spherical geometries.

Findings

Generalized isoperimetric inequalities for spherical and hyperbolic wave fronts.

Proved hyperbolic and spherical versions of Menzin's conjecture.

Extended bicycle curve concepts to curved Riemannian surfaces.

Abstract

We prove generalizations of the isoperimetric inequality for both spherical and hyperbolic wave fronts (i.e. piecewise smooth curves which may have cusps). We then discuss "bicycle curves" using the generalized isoperimetric inequalities. The euclidean model of a bicycle is a unit segment AB that can move so that it remains tangent to the trajectory of point A (the rear wheel is fixed on the bicycle frame). We extend this definition to a general Riemannian manifold, and concern ourselves in particular with bicycle curves in the hyperbolic plane H^2 and on the sphere S^2. We prove results along the lines of those proven by Levi and Tabachnikov and resolve both spherical and hyperbolic versions of Menzin's conjecture, which relates the area bounded by a curve to its associated monodromy map.