On relation between rational and differential rational invariants of surfaces with respect to the motion groups

TL;DR

This paper explores the relationship between rational and differential rational invariants of surfaces under motion groups, proposing a method that extends beyond traditional moving frame techniques in differential geometry.

Contribution

It introduces a new approach to describe surface invariants using a commuting system of invariant derivatives and a finite set of invariants, applicable in broader contexts.

Findings

Existence of a commuting system of invariant derivatives.

Finite generating set of invariants for surfaces.

Method extends the applicability of invariant description methods.

Abstract

The description of invariants of surfaces with respect to the motion groups is reduced to the description of invariants of parameterized surfaces with respect to the motion groups. Existence of a commuting system of invariant partial differential operators (derivatives) and a finite system of invariants, such that any invariant of the surface is a function of these invariants and their invariant derivatives, is shown. The offered method is applicable in more general settings than the "Moving Frame Method" does in differential geometry.