# Characterizing a surface by invariants

## TL;DR

This paper introduces canonical principal parameters for surfaces in three-dimensional space without umbilical points, showing that such surfaces are uniquely determined by a pair of invariants satisfying a specific PDE, which can be chosen as principal curvatures or Gauss and mean curvature.

## Contribution

It establishes a new canonical parametrization for surfaces and proves their unique determination by invariants satisfying the Gauss equation.

## Key findings

- Surfaces without umbilical points can be characterized by invariants in canonical parameters.
- The invariants satisfy a PDE equivalent to the Gauss equation.
- Principal curvatures or Gauss and mean curvature can serve as these 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.