Etude des (n+1)-tissus de courbes en dimension n

TL;DR

This paper generalizes the concept of Blaschke curvature to (n+1)-webs of curves in n-dimensional manifolds, classifies rank-one webs in dimension three, and provides methods to distinguish different classes using invariants.

Contribution

It introduces a generalized curvature for higher-dimensional webs, classifies rank-one 4-webs in dimension three, and develops invariants to distinguish their isomorphism classes.

Findings

Generalized Blaschke curvature for (n+1)-webs of curves.

Existence of infinitely many isomorphism classes of rank-one 4-webs in dimension three.

Development of invariants to distinguish web classes, including quadrilateral webs.

Abstract

For $(n+1)$-webs by curves in an ambiant $n$-dimensional manifold, we first define a generalization of the well known Blaschke curvature of the dimension two, which vanishes iff the web has the maximum possible rank which is one. But, contrary to the dimension two where all 3-webs of rank one are locally isomorphic, we prove that there are infinitely many classes of isomorphism for germs of 4-webs by curves of rank one in the dimension three : we provide a procedure for building all of them, up to isomorphism, and give examples of invariants of these classes allowing in particular to distinguish the so-called quadrilateral webs among them.