Conformal Geometry of One-parameter Families of Curves

A single regular analytic arc in the plane has no conformai differential invariants. The conformai theory of curvilinear angles was initiated by Kasner, and has been elaborated by him and others. The present paper is concerned with conformai differential invariants of a real one-parameter family of regular analytic arcs in the plane. We assume that the family is defined in some region R of the (x, y)-pla.ne by an equation of the form: u(x, y)= constant, where u is a singlevalued function which satisfies the conditions: (1) u is analytic in the region R, (2) u assumes real values for real values of x and y, (3) ul+ul does not vanish in R. By a conformai transformation we shall mean a real conformai transformation, nonsingular in R. Our principal results are : When a family u — c is transformed conformally into a family U = c, the parameters of the two families being the same, the quantity A = (uxx+Uyy)/(ul+uy), and certain conformally invariant derivatives of A are unaltered. There exist rational functions of A and these derivatives which are independent of the parameter in terms of which the family u = constant is expressed. We obtain a geometric interpretation of the invariants and apply the results to a generalization of isothermal families.