2025/12/22 by Altavilla, Amedeo, Schröcker, Hans-Peter, Šír, Zbyněk +1
#53A30 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG) #Primary 65D17 #Secondary 30C35
paper · doi:10.48550/arxiv.2512.19587
We fully characterize the mappings Φ that send every Pythagorean-hodograph (PH) curve to a PH curve. We prove that in any dimension, such mappings are precisely the conformal functions whose dilation is the square of a real rational function. In the planar case, this implies (up to conjugation) that ∂Φ/∂ z = Ψ2, where Ψ is meromorphic and satisfies Res(Ψ2) = 0 at every pole. In higher dimensions, PH preservation forces Φ to be a conformal map; for n ≥ 3, Liouville's theorem then implies that any local diffeomorphism with this property is (anti-)Möbius. These results subsume the previously known ``(scaled) PH-preserving'' constructions of mappings ℝ2 → ℝ3 and align with Ueda's conformal viewpoint on isothermal and spherical geometries. At the level of examples, we demonstrate how PH-preserving mappings relate to the construction of rational PH curves and minimal surfaces.