It is not unusual to consider on
a surface a conformal structure determined by a positive definite quadratic form
which may or may not be the official Riemannian metric on the surface. Given a
smooth mapping with positive Jacobian between a pair of surfaces each provided
with such a conformal structure, we describe in this paper an obvious procedure for
computing the dilatation of the mapping. Next, we consider surfaces smoothly
immersed in Ea, and mappings (called allowable) for which dilatation is a
function of the principal curvatures at corresponding points. Referring to a
conformal structure as geometrically significant if determined by a linear
combination of the fundamental forms with coefficients which are smooth functions
of the principal curvatures, we show (for example) that a mapping which
preserves lines of curvature is allowable between any pair of geometrically
significant conformal structures if it is allowable between any one pair of
geometrically significant conformal structures. Finally, we prove that a complete
surface smoothly immersed in E3 on which K ≦ 0 and H2− K ≡ c≠0 is
conformally equivalent either to the finite plane or to the once punctured finite
plane.
