Vol. 22, No. 1, 1967

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The dilatation of some standard mappings

Tilla Weinstein

Vol. 22 (1967), No. 1, 117–124

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 c0 is conformally equivalent either to the finite plane or to the once punctured finite plane.

Mathematical Subject Classification
Primary: 53.25
Received: 18 July 1966
Published: 1 July 1967
Tilla Weinstein