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

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).