Conformal classification of analytic arcs or elements: Poincaré’s local problem of conformal geometry

Abstract

In the geometry based on the infinite group of conformal transformations of the plane (or on the equivalent theory of analytic functions of one complex variable), two types of problems must be carefully distinguished: those relating to regions and those relating to curves or arcs. Two regions of the plane are equivalent when there exists a conformal representation of the one on the other, the representation to be regular at every interior point. The classic Riemann theory shows that all simply connected regions are equivalent, any one being convertible into say the unit circle. The difficulties connected with the behavior of the boundary (which may be a Jordan curve or a more general point set) have been cleared up in the recent papers of Osgood, Study, and Caratheodory, Logically simpler problems relating to curves or arcs have received very scant attention. Two arcs are equivalent provided the one can be converted into the other by a conformal transformation, the transformation to be regular at the points of the arcs, and therefore in some (unspecified) regions including the arcs in their interiors. The main problem hitherto discussed by the writer in his papers on conformal geometry is the invariant theory of curvilinear angles.t Such a configuration (which may be designated also as an analytic angle) consists of two arcs through a common point, both arcs being real, analvtic, and regular at the vertex.: In this theory it is necessary to distinguish rational and irrational angles. If 0 denotes the magnitude of the angle (invariant of first order), then when 0/uis rational there exists a unique conformal invariant