Les douze surfaces de Darboux et la trialit{\'e}

TL;DR

This paper provides a geometric interpretation of Darboux's twelve surfaces, linking them to a differential-geometric version of triality involving isotropic surfaces in a 6D quadric.

Contribution

It introduces a new geometric perspective on Darboux's twelve surfaces through the lens of triality and isotropic surface theory in a neutral signature quadric.

Findings

Interpretation of Darboux's twelve surfaces as triality-related geometric objects

Connection between surface deformations and isotropic surfaces in a 6D quadric

Insight into the structure of infinitesimal isometric deformations

Abstract

The purpose of this article is to give a geometric interpretation to the so-called "twelve surfaces of Darboux", or "Darboux wreath", which appear by applying repeatedly certain simple transformations to a given infinitesimal isometric deformation of a surface in euclidean three space. This interpretation is a differential-geometric version of triality, concerning totally isotropic surfaces immersed in the real 6-dimensional projective quadric defined by a quadratic form of neutral signature (4,4).