On Differential Invariants of Parabolic Surfaces

TL;DR

This paper demonstrates that the algebra of differential invariants for generic parabolic surfaces under a specific group is generated by a single fifth-order invariant, simplifying the understanding of their geometric properties.

Contribution

It establishes that all differential invariants of these surfaces can be derived from one fifth-order invariant, using Fels-Olver's recurrence formulas, revealing a minimal generating set.

Findings

The algebra of invariants is generated by one invariant of order 5.

The invariant $M$ has 57 differential monomials.

The proof uses Fels-Olver's recurrence formulas on jet bundles.

Abstract

The algebra of differential invariants under $SA_3(\mathbb{R})$ of generic parabolic surfaces $S^2 \subset \mathbb{R}^3$ with nonvanishing Pocchiola $4^{\text{th}}$ invariant $W$ is shown to be generated, through invariant differentiations, by only one other invariant, $M$, of order $5$, having $57$ differential monomials. The proof is based on Fels-Olver's recurrence formulas, pulled back to the parabolic jet bundles.