On the distortion of conformal immersions between 2-space forms

Abstract

Let 0~ and ~1 be 2-dimensional space forms (connected and complete Riemannian manifolds of class C 3 and of constant curvature K). We normalize K = 1, 0, 1, and speak of spherical, Euclidean, and hyperbolic space forms. Denote by O and 91 the metric tensors of ~ and ~ t . If f : ~ / -*~ is a conformal immersion and f , the induced linear map of the tangent bundles, then the distortion of f is the map