A step to Gronwall's conjecture

TL;DR

This paper proposes a new approach to prove Gronwall's conjecture by introducing an invariant called the characteristic, which helps identify when a local isomorphism between plane linear 3-webs is a homography.

Contribution

It introduces the characteristic invariant for generic points of linear 3-webs, advancing the understanding of conditions under which web isomorphisms are homographies.

Findings

The characteristic invariant distinguishes when a web isomorphism is a homography.

A new method to approach Gronwall's conjecture using invariants.

Potential progress towards proving Gronwall's conjecture.

Abstract

In this paper we will explore a way to prove the hundred years old Gronwall's conjecture: if two plane linear 3-webs with non-zero curvature are locally isomorphic, then the isomorphism is a homography. Using recent results of S. I. Agafonov, we exhibit an invariant, the {\sl characteristic}, attached to each generic point of such a web, with the following property: if a diffeomorphism interchanges two such linear webs, sending a point of the first to a point of the second which have the same characteristic, then this diffeomomorphism is locally a homography.

Equations

$${\partial }_x{\partial }_y(\log \frac{{\partial }_{x}f}{{\partial }_{y}f})dx\wedge dy.$$

$$\Pi =(P-Q)(Q-R)(R-P),$$

$$\Delta =(P-Q)R_y+(Q-R)P_y+(R-P)Q_y,$$

$$ca{r}_W:=\Pi .(P_{yy}+Q_{yy}+R_{yy})/{\Delta }^2.$$

$$ca{r}_W(M)=ca{r}_{\bar{W}}(\psi (M)),$$

$$ca{r}_W(M)=ca{r}_{\bar{W}}(\varphi (M))$$

$$M=(0,0),A_M=(1,1),B_M=(1,-1),C_M=(2,0).$$

$$a(t)=1-t+a_2{t}^2+\cdots +a_i{t}^i+\cdots ,$$

$$b(t)=-1-t+b_2{t}^2+\cdots +b_i{t}^i+\cdots ,$$

$$c(t)=-t/2+c_2{t}^2+\cdots +c_i{t}^i+\cdots .$$

$$ca{r}_W(0,0)=4(a_2+b_2+c_2).$$

$$1+u-v+(1+a_2)u^2+(-1-2a_2)vu+a_2{v}^2,$$

$$-1-u-v+(-1+b_2)u^2+(-1+2b_2)vu+b_2{v}^2$$

$$-v/2-uv/4+c_2{v}^2.$$

$$a_2+b_2+c_2={\bar{a}}_2+{\bar{b}}_2+{\bar{c}}_2,$$

$$det\left|\matrix{1&1&1\cr x&y&z\cr a(x)&b(y)&c(z)\cr}\right|=0\ .$$

$$F_{\mu }(x,y)=x+y+xy(x-y)(\mu +g(x,y))$$

$$h(x,y)=x+y+xy(x-y)k(x,y)$$

$$F_{\mu }(x,y)=x+y+xy(x-y)(\mu +g_{\mu }(x,y)),$$

$${\bar{F}}_{\mu }(x,y)=x+y+xy(x-y)(\mu +{\bar{g}}_{\mu }(x,y)).$$

$$g_{\mu }={\bar{g}}_{\mu }.$$

$$\mu =a_2+b_2+c_2$$

$${\bar{a}}_2+{\bar{b}}_2+{\bar{c}}_2=\mu .$$

$$E_{10}=(-2a_2-2b_2+c_2+20a_3+8b_3+14c_3)/7.$$

$$E_{01}=(2a_2+2b_2-c_2+20b_3+8a_3+14c_3)/7.$$

$$E_{20}=a_2/12+b_2/12-c_2/6+10a_2^2/3+c_2^2+2a_2{b}_2/3-2b_2{c}_2/3+2a_2{c}_2/3$$

$$-2a_3+c_3/3+20a_4/3+4b_4/3+3c_4.$$

$$E_{11}=-a_2/12-b_2/12-c_2/3+a_2^2/3-b_2^2/3+4c_2(a_2-b_2)/3+4a_4+4b_4+5c_4.$$

$$E_{02}=a_2/12+b_2/12-c_2/6-10b_2^2/3-c_2^2-2a_2{b}_2/3-2b_2{c}_2/3+2a_2{c}_2/3$$

$$+2b_3-c_3/3+20b_4/3+4a_4/3+3c_4.$$

Full text

Using recent results of S. I. Agafonov, we exhibit an invariant, the characteristic, attached to each generic point of such a web, with the following property: if a diffeomorphism interchanges two such linear webs, sending a point of the first to a point of the second which have the same characteristic, then this diffeomomorphism is locally a homography.

Keywords: planar 3-webs. Gronwall conjecture.

AMS classification : 53A60

1 Introduction

In this text we work in the real projective plane. The results are the same in the complex case. We also work in the analytic case.

A plane 3-web is a triple of 1-dimensional foliations, two by two tranversal, on an open domain of the plane. In the sequel we will work only with these 3-webs : so we forgot the word plane. Such a 3-web is called linear if the leaves of the foliations are rectilinear.

Let $W$ and $\bar{W}$ be two 3-webs, linear or not, defined respectively on domains $U$ and $\bar{U}.$ We say that they are isomorphic if there is a diffeomorphism from $U$ to $\bar{U}$ which maps every leaf of the foliations of $W$ on a leaf of the foliations of $\bar{W}.$

Near any point of the domain of a 3-web $W,$ there are coordinates $(x,y)$ (in general non affine) such that the foliations are given by the verticals $x={constant},$ the horizontals $y={constant}$ and the level sets of some function $f$ (the sets of $(x,y)$ such that $f(x,y)={constant}).$ Such a 3-web is denoted $(x,y,f).$ In other words we can say that $W$ is locally isomorphic to some $(x,y,f).$

Attached to each 3-web there is a 2-form called its (Blaschke) curvature. For $(x,y,f)$ this curvature is

[TABLE]

We say that a 3-web is non flat if its curvature doesn’t vanish at any point of its domain.

**Gronwall’s conjecture : ** If two non flat linear 3-webs are isomorphic, then the diffeomorphism which realizes this isomorphism is a homography near any point.

Probably the recent paper of S.I. Agafonov ([SA]) contains the best historical references on this subject. We will also use some notions appearing in this paper and, particularly, the following.

We choose affine coordinates $(x,y)$ on an open subset $U$ of the plane. Let $W$ a linear 3-web defined on $U$ by the slopes $P,$ $Q$ and $R$ of the different foliations ($P(x,y)$ is the slope of the first foliation at the point of coordinates $(x,y)$ …). We define following quantities

[TABLE]

[TABLE]

with the convention: if $f$ is a function of $(x,y)$ we denote $f_{y}$ its derivative with respect to $y,$ $f_{yy}$ its second derivative with respect to $y$…. We assume now that $W$ is non flat. The Lemma 1 of [SA] implies that we can assume also that $\Delta$ is everywhere non vanishing.

Définition 1.1

The characteristic of $W$ is

[TABLE]

In [SA] we can find a complete set of projective invariants (invariant up to homographies) and $car_{W}$ is the sum of three of them.

Because $car_{W}$ is a projective invariant, if a homography $\psi$ interchanges linear 3-webs $W$ and $\bar{W}$, we have

[TABLE]

for any point $M.$

Our central result is the following.

Theorem 1.2

We consider two non flat linear 3-webs $W$ and $\bar{W}.$ We assume that there is an isomorphism $\phi$ from the first one to the second. If there is a point $M$ such that

[TABLE]

then the isomorphism is a homography near $M.$

This theorem says that Gronwall’s conjecture is true if we can always find a point $M$ satisfying condition (1). We can hope to find such $M$ by a fix point method.

2 Description of linear 3-webs near a point.

We consider a linear 3-web $W,$ and $M$ a point of its domain. Then we can prolongate the three foliations to obtain three families of lines $\cal A,$ $\cal B$ and $\cal C.$ In the projective plane $\cal A,$ $\cal B$ and $\cal C$ envelop three curves, which may degenerate into a point. We denote respectively $A_{M},$ $B_{M}$ and $C_{M}$ the focal points on the three line passing by $M$ (the points where these lines touch the envelops). We impose now that $W$ is non flat. Then the Lemma 1 of [SA] implies also that $(M,A_{M},B_{M},C_{M})$ is a projective frame. Up to a homography, we can choose coordinates $(u,v)$ such that

[TABLE]

Near the origin each leaf of our foliations is transversal to the $v$-axis. So our 3-web can be described as follows.

There are three 1-variable functions $a:x\mapsto a(x)$ (resp. $b:y\mapsto b(y)$, $c:z\mapsto c(z)$) such $\cal A$ (resp $\cal B$, $\cal C$) consists of the lines $v=a(x)u+x$ (resp. $v=b(y)u+y$, $v=c(z)u+z$) where $x$ (resp. $y$, $z$) is a parameter varying near the origin.

Be aware that the functions $a$, $b$ and $c$ we just defined aren’t the $a$, $b$ and $c$ of [SA].

Because of the choice of $(u,v)$, Taylor expansions of $a,$ $b$ and $c$ have the shapes

[TABLE]

[TABLE]

[TABLE]

Lemme 2.1

The following formula holds:

[TABLE]

Moreover $a_{2}+b_{2}+c_{2}$ vanishes if and only if the curvature of $W$ at the origin vanishes.

To prove this lemma we begin to compute the 2-order Taylor expansion of the function $x(u,v)$ (resp. $y(u,v),$ resp. $z(u,v)$) given by the implicit relation $v=a(x(u,v))u+x(u,v)$ (resp. $v=b(y(u,v))u+y(u,v),$ resp. $v=c(z(u,v))u+z(u,v)$). We find they are $-u+v-u^{2}+uv$ for $x(u,v),$ $u+v+u^{2}+uv$ for $y(u,v)$ and $v+uv/2$ for $z(u,v)$).

Then the slope functions are $P(u,v)=a(x(u,v)),$ $Q(u,v)=b(y(u,v))$ and $R(u,v)=c(z(u,v)).$ So their 2-order Taylor expansions are respectively

[TABLE]

[TABLE]

and

[TABLE]

So the values of $\Pi,$ $\Delta$, $P_{yy}$, $Q_{yy}$ and $R_{yy}$ at the origin are respectively $2,$ $1,$ $2a_{2},$ $2b_{2}$ and $2c_{2}.$ This proves the first assertion of the lemma.

The second can be proved, for example, using the formula of the curvature given in the introduction.

3 Isomorphic linear webs.

In order to prove theorem 1.2, we consider two non flat linear 3-webs $W$ and $\bar{W},$ the first near a point $M$, the second near a point $\bar{M}.$ For both we adopt a description as in the preceeding section : the first is described by the three local functions $a,$ $b$ and $c,$ the second is described by the local functions $\bar{a},$ $\bar{b}$ and $\bar{c}.$ Via the lemma 2.1, the relation $car_{W}(M)=car_{\bar{W}}(\bar{M})$ writes as

[TABLE]

with evident notations.

In the following of this section we propose a way to express the existence of an isomorphism from $W$ to $\bar{W}$ mapping $M$ to $\bar{M}.$

Let $x,$ $y,$ $z$ be three numbers such that the three leaves $v=a(x)u+x,$ $v=b(y)u+y,$ $v=c(z)u+z$ of $W$ have a commun point. They are characterized by the relation

[TABLE]

This equation defines implicitly a function $z=f_{W}(x,y)$ on a neigborhood of the origin. This proves that $W$ is locally isomorphic to $(x,y,f_{W}).$ Because the change of coordinates $(u,v)\mapsto(x,y)$ is not a homography (in general), $(x,y,f_{W})$ has no reason to be linear.

Replacing respectively $a$ with $\bar{a},$ $b$ with $\bar{b},$ $c$ with $\bar{c},$ we can construct $f_{\bar{W}}$ such that $\bar{W}$ is isomorphic to the 3-web $(x,y,f_{\bar{W}}).$

Now $W$ is locally isomorphic to $\bar{W}$ by a diffeomorphism which maps $M$ to $\bar{M},$ if and only if there is a local isomorphism, preserving the origin, from $(x,y,f_{W})$ to $(x,y,f_{\bar{W}}).$

Proposition 3.1

Let $\mu$ be any non zero number. Every 3-web $(x,y,f),$ with non-zero curvature at the origin, is locally isomorphic to an unique 3-web $(x,y,F_{\mu})$ such that

[TABLE]

where $g$ is a function vanishing at the origin.

This proposition is a particular case of the existence and unicity of normal form for 3-webs which appears for the first time in [DJ]: for any 3-web $V$ near any point $m,$ there are local coordinates $(x,y)$ vanishing at $m$ such that $V$ becomes $(x,y,h)$ with

[TABLE]

where $k$ may be any function. Moreover $k$ is unique up to a homothety $(x,y)\mapsto(\lambda x,\lambda y).$ For the moment we impose no assumption concerning curvature. If the curvature doesn’t vanish at $m$ then $k(0,0)$ is different from zero. Then, up to a homothety, we can assume $k(0,0)=\mu$ and we obtain the above proposition.

The fact that $W$ and $\bar{W}$ are non flat implies that $(x,y,f_{W})$ and $(x,y,f_{\bar{W}})$ have non zero curvature at the origin. The above proposition says that there are two functions $g_{\mu}$ and $\bar{g}_{\mu}$, vanishing at the origin, such that $W$ and $\bar{W}$ are respectively isomorphics to $(x,y,F_{\mu})$ and $(x,y,\bar{F}_{\mu}),$ with

[TABLE]

[TABLE]

The unicity part of the above proposition implies the following lemma.

Lemme 3.2

Fix the non zero number $\mu.$ Then $W$ is isomophic to $\bar{W},$ by an isomorphism which maps $M$ to $\bar{M},$ if and only if

[TABLE]

To be able to express the isomorphy of $W$ and $\bar{W}$ with this lemma, we need to have a practical method to compute $g_{\mu}$ and $\bar{g}_{\mu}.$

We don’t have such a method but, at least, we will give in the next section an algorithm which computes the $k$-jet of $g_{\mu}$ (resp. $\bar{g}_{\mu}$), starting with $(k+2)$-jets of $a$, $b$ and $c$ (resp. $\bar{a}$, $\bar{b}$ and $\bar{c}$). If $W$ and $\bar{W}$ are isomorphic and for every $k$, this will gives many precise polynomial equations between $(a_{2},b_{2},c_{2},\cdots,a_{k+2},b_{k+2},c_{k+2})$ and $({\bar{a}}_{2},{\bar{b}}_{2},{\bar{c}}_{2},\cdots,{\bar{a}}_{k+2},{\bar{b}}_{k+2},{\bar{c}}_{k+2}).$

4 Algorithm to compute $k$-jets of $g_{\mu}.$

In this section we mimic a classical proof of the existence of normal form for any 3-web to get an algorithm which works for jets.

We choose

[TABLE]

i.e. $\mu$ is the characteristic of $W$ at the origin up to the factor 4.

The input is the $(k+2)$-jet of $a$, $b$ and $c$, i.e. the numbers $a_{2},\cdots,a_{(k+2)},$ $b_{2},\cdots,b_{(k+2)}$ and $c_{2},\cdots,c_{(k+2)}.$

The first procedure gives the $(k+3)$-jet, $j^{(k+3)}f_{W}$ of $f_{W}$ at the origin by computing the $(k+3)$-jet of the solution of the implicit relation (2) when we replace respectively $a,$ $b$ and $c$ by their $(k+2)$-jets.

The second procedure is the normalisation of $j^{(k+3)}f_{W}$ to obtain the $k$-jet of $g_{\mu}.$ We do that in six steps.

First step. We use the simplifying notation $F=j^{(k+3)}f_{W}$ and consider it as a polynomial function with two variables. Compute $X,$ the $(k+3)$-jet of the inverse function of $t\mapsto F(t,0).$

Second step. Compute $Y,$ the $(k+3)$-jet of the inverse function of $t\mapsto F(0,t).$

Third step. Compute the $(k+3)$-jet of $F(X(x),Y(y)).$ We denote it by $G$ (remark that $G$ has the shape $x+y+xy\Theta(x,y)$)

Fourth step. Consider $K=G(t,t).$ Find the 1-variable polynomial $U=t+u_{2}t^{2}+\cdots u_{(k+3^{)}}t^{(k+3)}$ such that the $(k+3)$-jet of $G(U(t),U(t))$ is egal to $U(2t)$ (it exists by the classical Sternberg’s theorem [SS] which says that every map $t\mapsto 2t+d_{2}t^{2}+\cdots$ is conjugated to its linear part ; it is also unique because $j^{1}U=t$).

Fifth step. Compute $V,$ the $(k+3)$-jet of the inverse of $U,$ and $H,$ the $(k+3)$-jet of $V(G(U(x),U(y)))$ (remark that $H$ has the shape $x+y+xy(x-y)\Psi(x,y)$).

Sixth step. Compute $L,$ the $k$-jet of $(H-x-y)/(xy(x-y))$.

The output is $E=L-\mu$ which is the k-jet of $g_{\mu}.$

We have implemented this algorithm on Maple. It works very rapidly if $k$ is less or equal to seven.

5 Proof of Theorem 1.2.

We keep notation $\mu:=a_{2}+b_{2}+c_{2}.$ The assumption of our theorem writes as

[TABLE]

Using the algorithm of the preceeding section we compute respectively the 5-jets $E$ of $g_{\mu}$ and $\bar{E}$ of $\bar{g}_{\mu}.$

We write the Taylor expansion of $E$ as $E_{10}x+E_{01}y+\cdots+E_{ij}x^{i}y^{j}.$ We have

[TABLE]

[TABLE]

And also

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The following $E_{ij}$ may have very long expressions. For example if $i+j=5$ they contain nearly hundred terms. We only retain that they are polynomial expressions, with rational coefficients, in some of the $a_{r},$ $b_{r}$ and $c_{r}$ variables.

In the case of $\bar{E}$, we obtain the same expressions for its coefficients ${\bar{E}}_{ij}$ except we have to change respectivily $a_{r},$ $b_{r}$ and $c_{r}$ by $\bar{a}_{r},$ $\bar{b}_{r}$ and $\bar{c}_{r}.$

To simplify we use the following notations:

[TABLE]

for every $r.$ The hypothesis of our theorem writes as $C_{2}=-A_{2}-B_{2}.$ To prove it we have to prove $A_{r}=B_{r}=C_{r}=0$ for every $r.$

We adopt notations

[TABLE]

for every $i$ and $j.$ The existence of an isomorphism between $W$ and $\bar{W}$ implies the set of equations $T_{ij}=0.$ They are polynomial equations with unknown $A_{r},$ $B_{r}$ and $C_{r}$ and coefficients rational in some of the $a_{r},$ $b_{r},$ $c_{r}.$

For example the relations (3) and (4) give $T_{10}$ and $T_{01},$ i.e. the order one equations. Equations $T_{10}=0$ and $T_{01}=0,$ give relations

[TABLE]

[TABLE]

At order 2 the equations $T_{20}=0,$ $T_{11}=0$ and $T_{02}=0$ give

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Note that the second members of these relations are polynomial with variables $A_{2},$ $B_{2},$ $C_{3},$ $a_{2},$ $b_{2}$ and $c_{2}.$

At order 3 and using Maple, we obtain four equations $T_{ij}=0$ and they allow to express $A_{5},$ $B_{5},$ $C_{5}$ and $C_{3}$ as rational fonctions of the variables $A_{2},$ $B_{2},$ $a_{2},$ $b_{2},$ $c_{2},$ $a_{3},$ $b_{3}$ and $c_{3},$ with the denominator $a_{2}+b_{2}+c_{2}.$ Formulas are two long to be reproduced here.

For the moment we skip order 4 and consider the six equations $T_{ij}=0$ with $i+j=5.$ Maple proves that they allow to express $A_{6},$ $B_{6},$ $C_{6},$ $A_{7},$ $B_{7}$ and $C_{7}$ rationally in function of $A_{2},$ $B_{2},$ $a_{2},$ $b_{2},$ $c_{2},$ $\cdots,$ $a_{5},$ $b_{5}$ and $c_{5}$ with the denominator $(a_{2}+b_{2}+c_{2})^{2}.$

We remark that $A_{2}=B_{2}=0$ implies that $A_{i},$ $B_{i}$ and $C_{i}$ vanish for $i=3,\cdots,7.$

Now we compute the five equations $T_{ij}=0$ with $i+j=4.$ They have the shape

[TABLE]

where $\alpha_{j},\beta_{j},\gamma_{j},\mu_{j},\nu_{j}$ are rationally functions of $b_{2},$ $c_{2},$ $\cdots,$ $a_{5},$ $b_{5}$ and $c_{5}$ with the denominator $a_{2}+b_{2}+c_{2}.$

We see also that the $(3\times 5)$-matrix with lines $(\alpha_{j},\beta_{j},\gamma_{j})$ has rank 3 for any value of $b_{2},$ $c_{2},$ $\cdots,$ $a_{5},$ $b_{5}$ and $c_{5}.$ This is a consequence of the fact that the coefficients of this matrix depends only on the three numbers $a_{2}+b_{2}+c_{2},$ $a_{3}+b_{3}+c_{3}$ and $a_{2}+b_{2}+2b_{3}-2a_{3}.$ So our system of equations can be rewriten as

[TABLE]

To obtain the Maple worksheet which gives these results, contact the author at dufourh@netcourrier.com

Lemme 5.1

For $i+j>2$ we have relations

[TABLE]

where $\theta_{ij},$ $\phi_{ij}$ and $\psi_{ij}$ are some constants and $S_{ij}$ is a polynomial with variables $a_{2},$ $b_{2},$ $c_{2},$ $\cdots,$ $a_{i+j+1},$ $b_{i+j+1}$ and $c_{i+j+1}.$ Moreover any $(3\times 3)$-submatrix of the matrix whose lines are $(\theta_{ij},\phi_{ij},\psi_{ij})$ is of rank 3.

This can be proven for any $(i,j)$ (whithout Maple !) as follows. We use notation $n=i+j+2.$ We first see that $F,$ the $(n+1)$-jet of $f_{W},$ has the shape

[TABLE]

where $K,$ $L$ and $M$ are homogeneous polynomials of degree $(n+1),$ with variable $(x,y)$ and constant coefficients; $\Theta$ is a polynomial expression with variables $x,$ $y,$ $a_{2},$ $b_{2},$ $c_{2},$ $\cdots,$ $a_{n-1},$ $b_{n-1}$ and $c_{n-1}.$ Now we apply the normalising procedure, described in the previous section, to $F.$ We only have to follow what happens to the terms containing $a_{n},$ $b_{n}$ and $c_{n}.$ It is a little long but elementary.

Note that, with this lemma, we recover in part above Maple results: we recover that the equations $T_{ij}=0$ for $i+j=n-2$ give expressions of $A_{n},$ $B_{n}$ and $C_{n}$ in terms of the previous $A_{p},$ $B_{p}$ and $C_{p},$ and some of the $a_{r}$, $b_{r}$ and $c_{r},$ for $n=4,5,6,7.$

This lemma proves also that, if $A_{2},$ $B_{2},$ $A_{3},$ $B_{3}$ and $C_{3}$ vanish then all the $A_{r},$ $B_{r}$ and $C_{r}$ vanish also. Using Maple calculations above we see that the relation $A_{2}=B_{2}=0$ implies that all the $A_{r},$ $B_{r}$ and $C_{r}$ vanish but also all the $T_{ij}.$

Using the three first equations (5) we can replace any monomial in $A_{2}$ and $B_{2}$ by a linear expression $\rho A_{2}+\tau B_{2}.$ So our system of equations $T_{ij}$ can rewriten as a set of equations

[TABLE]

for $r>2$ and an infinity of $k;$ the coefficients $u^{s}_{r},$ $v^{s}_{r},$ $w^{s}_{r}$ and $t^{s}_{r}$ depending only on the $a_{n},$ $b_{n}$ and $c_{n}.$

If this system has a non zero solution $(A_{2},B_{2},A_{3},B_{3},C_{3},\dots)$ then it has an infinity of solutions: $t(A_{2},B_{2},A_{3},B_{3},C_{3},\dots)$ for any number $t.$

This means that $W$ would be isomorphic to any linear 3-web $\bar{W}_{t}$ which is described by the three one variable functions

[TABLE]

where $A,$ $B$ and $C$ are the functions which have respectively Taylor expansions $A_{2}x^{2}+\cdots+A_{n}x^{n}+\cdots,$ $B_{2}y^{2}+\cdots+b_{n}y^{n}+\cdots$ and $C_{2}z^{2}+\cdots+C_{n}z^{n}+\cdots.$

This contredicts the known fact that any non flat linear 3-web can only be isomorphic to a finite number of homographically different linear 3-webs.

So we have proven that the only possiblity is $A_{r}=B_{r}=C_{r}=0$ for any $r$ and our Theorem.

6 Remarks.

Let $\phi$ be an isomorphism between the two linear non flat webs $W$ and $\bar{W}.$ We suppose also that $\phi$ maps a point $M$ of the domain of $W$ to a point $\bar{M}$ on the domain of $\bar{W}.$

We adopt the description of $W$ (resp. $\bar{W}$) near $M$ (resp. $\bar{M}$) of the section 2, i.e. with three 1-variable functions $a,$ $b$ and $c$ (resp. $\bar{a},$ $\bar{b}$ and $\bar{c}$). Then $\phi$ becomes a local diffeomorphism which preserves the origin. As it preserves the $u$-axis and the two bissectrices $u=v$ , $u=-v,$ its 1-jet at the origin is a homothety $kI$. Then we have

[TABLE]

For any linear 3-web we find in [SA] the construction of 1-forms $U_{1},$ $U_{2}$ and $U_{3}$ which are invariant up to homographies, such that the three foliations are given by the kernels of these forms and

[TABLE]

We denote $U_{1},$ $U_{2}$ and $U_{3}$ these forms for $W$ and $\bar{U}_{1},$ $\bar{U}_{2}$ and $\bar{U}_{3}$ for $\bar{W}.$

Classically, there is a function $f$ such that

[TABLE]

for every $i=1,2,3.$ Using the description of section 2, we can show

[TABLE]

So our result can be rephrased as: if $f^{2}$ is equal to 1 at some point then, near this point, $\phi$ is a homography.