Courbure des tissus planaires d\'efinis implicitement par une \'equation diff\'erentielle polynomiale en y'. Programmation

TL;DR

This paper develops a Maple program to compute the curvature of planar d-webs defined by polynomial differential equations and proves a concentration theorem for their curvature matrices in higher dimensions.

Contribution

It introduces a computational method for curvature of implicit planar d-webs and proves a new concentration theorem for curvature matrices in calibrated webs.

Findings

Maple program for curvature computation of d-webs

Proof of a concentration theorem for curvature matrices

Application to planar and higher-dimensional webs

Abstract

The aim of this paper is mainly, after some theoretical explanations, to provide a program on Maple for computing, whatever be d, the curvature of the planar d-web implicitely defined by a differential equation F(x,y,y')=0, F being polynomial of degree d with respect to y'. Moreover, we prove in the appendix a "concentration theorem" for any calibrated ordinary $d$-web of codimension one in n-dimensional manifold (in particular for any planar web). Its curvature matrix, relatively to an "adapted" trivialization, is concentrated on the (n-2+k0)!/(n-2)!k0! last lines (the last line if n=2), k0 denoting the integer such that d=(n-1+k0)!/(n-1)!(k0)!$.