The classical isoperimetric inequality relates the lengths of curves to the areas that they bound. More specifically, we have that for a smooth, simple closed curve of length L bounding area A on a surface of constant curvature c,
L2 ≥ 4πA – cA2
with equality holding only if the curve is a geodesic circle. 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), as discussed in [Finn, The College Mathematics Journal, 33, 2002], [Tabachnikov, Israel J. Math. 151: 1–28, 2006], and [Levi, Tabachnikov, Experiment. Math. 18: 173–186, 2009]. We extend this definition to a general Riemannian manifold, and concern ourselves in particular with bicycle curves in the hyperbolic plane H2 and on the sphere S2. We prove results along the lines of those in [Levi, Tabachnikov, Experiment. Math. 18: 173–186, 2009] and resolve both spherical and hyperbolic versions of Menzin's conjecture, which relates the area bounded by a curve to its associated monodromy map.
© de Gruyter 2011