Characterizing a surface by invariants

Abstract

Canonical principal parameters are introduced for surfaces in ${\mathbb{R}}^3$ without umbilical points. It is proved that in these parameters the surface is determined (up to position in space) by a pair of invariants satisfying a partial differential equation equivalent to the Gauss equation. As such a pair of invariants we may use the principal curvatures or the Gauss and the mean curvature.

8 pages

References in corpus (2)

• Canonical Weierstrass Representation of Minimal Surfaces in Euclidean Space
• Explicit Solving of the System of Natural PDE's of Minimal Surfaces in the Four-Dimensional Euclidean Space