Mehdi Belraouti

Mohamed Deffaf

Yazid Raffed

Abdelghani Zeghib

Pseudo-Conformal

PSEUDO-CONFORMAL ACTIONS OF THE MÖBIUS GROUP

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

A pseudo-Riemannian manifold is a differentiable manifold M endowed with a pseudo-Riemannian metric g of signature (p, q). Two metrics g 1 and g 2 on M are said to be conformally equivalent if and only if g 1 = exp(f )g 2 where f is C ∞ function. A conformal structure is then an equivalence class [g] of a pseudo-Riemannian metric g and a conformal manifold is a manifold endowed with a pseudo-Riemannian conformal structure. A remarkable family of conformal manifolds is given by the conformally flat ones. These are pseudo-Riemannian conformal manifolds that are locally conformally diffeomorphic (i.e preserving the conformal structures) to the Minkowski space R p,q i.e the vector space R p+q endowed with the pseudo-Riemannian metric -dx 2 0 -... -dx 2 p-1 + dy 2 0 + ... + dy 2 q-1 . The conformal group Conf(M, g) is the group of transformations that preserve the conformal structure [g]. It is said to be essential if there is no metric in the conformal class of g for which it acts isometrically. In the Riemannian case, the sphere S n is a compact conformally flat manifold with an essential conformal group. The Einstein universe Ein p,q is the equivalent model of the standard conformal sphere in the pseudo-Riemannian setting. It admits a two-fold covering conformally equivalent to the product S p ×S q endowed with the conformal class of -g S p ⊕g S q . It is conformally flat and its conformal group, which is in fact the pseudo-Riemannian Möbius group O(p + 1, q + 1), is essential. Actually the Einstein universe is the flat model of conformal pseudo-Riemannian geometry. This is essentially due to the fact that the Minkowski space embeds conformally as a dense open subset of the Einstein universe Ein p,q and in addition to the Liouville theorem asserting that conformal local diffeomorphisms on Ein p,q are unique restrictions of elements of O(p + 1, q + 1). Hence a manifold is conformally flat if and only if it admits a (O(p + 1, q + 1), Ein p,q )-structure.

In the sixties A. Lichnérowicz conjectured that among compact Riemannian manifolds, the sphere is the only essential conformal structure. This was generalised and proved independently by Obatta and Ferrand (see [START_REF] Obata | The conjectures on conformal transformations of Riemannian manifolds[END_REF], [START_REF] Lelong-Ferrand | Transformations conformes et quasiconformes des variétés riemanniennes; application à la démonstration d'une conjecture de A. Lichnerowicz[END_REF]). In the pseudo-Riemannian case, a similar question, called the pseudo-Riemannian Lichnérowicz conjecture, was raised by D'Ambra and Gromov [START_REF] Ambra | Lectures on transformation groups: geometry and dynamics[END_REF]. Namely, if a compact pseudo-Riemannian conformal manifold is essential then it is conformally flat. This was disproved by Frances see [START_REF] Frances | About pseudo-Riemannian Lichnerowicz conjecture[END_REF], [START_REF] Frances | The lorentzian Lichnerowicz conjecture for real-analytic, three-dimensional manifolds[END_REF].

The present article is the first of a series on the pseudo-Riemannian Lichnérowicz conjecture in a homogeneous setting [START_REF] Belraouti | On homogeneous holomorphic conformal structures[END_REF][START_REF] Belraouti | Conformal pseudoriemannian Lichnerowicz conjecture in homogeneous setting[END_REF]. The general non homogeneous case, but with signature restrictions, was amply studied by Zimmer, Bader, Nevo, Frances, Zeghib, Melnick and Pecastaing (see [START_REF] Zimmer | Split rank and semisimple automorphism groups of G-structures[END_REF], [START_REF] Bader | Conformal actions of simple Lie groups on compact pseudo-Riemannian manifolds[END_REF], [START_REF] Frances | Some remarks on conformal pseudo-Riemannian actions of simple Lie groups[END_REF], [START_REF] Pecastaing | Conformal actions of real-rank 1 simple Lie groups on pseudo-Riemannian manifolds[END_REF], [START_REF] Pecastaing | Conformal actions of higher rank lattices on compact pseudo-Riemannian manifolds[END_REF], [START_REF] Pecastaing | Essential conformal actions of PSL(2, R) on real-analytic compact Lorentz manifolds[END_REF], [START_REF] Melnick | The conformal group of a compact simply connected Lorentzian manifold[END_REF]). Let us also quote [START_REF] Leistner | Conformal transformations of cahen-wallach spaces[END_REF] as a recent work in the Lorentz case.

We are investigating in this first part the case where the non-compact semi-simple part of the conformal group is locally isomorphic to the Möbius group SO(1, n + 1). More exactly, we prove the following classification result. This Möbius situation will actually play a central role towards the general case treated in [START_REF] Belraouti | On homogeneous holomorphic conformal structures[END_REF].

Theorem 1.1. Let (M, [g]) be a conformal connected compact pseudo-Riemannian manifold. We suppose that there exists G a subgroup of the conformal group Conf(M, g) acting essentially and transitively on (M, [g]). We suppose moreover that the noncompact semi-simple part of G is locally isomorphic to the Möbius group SO(1, n + 1). Then (M, [g]) is conformally flat. More precisely (M, [g]) is conformally equivalent to

• The conformal Riemannian n-sphere or;

• Up to a cover, the Einstein universe Ein 1,1 or; • Up to a finite cover, the Einstein universe Ein 3,3 . Remark 1.2. It turns out that, in the first and third cases, the acting group G is locally isomorphic to the Môbius group, that is, G is simple. In the second case, the universal cover G is a subgroup of SL(2, R) × SL(2, R). It can in particular be SL(2, R) × SO(2).

Preliminaries

2.1. Notations. Throughout this paper (M, g) will be a compact connected pseudo-Riemannian manifold of dimension n endowed with a transitive and essential action of the conformal group G = Conf(M, g). We suppose without loss of generality that G is connected.

Fix a point x in M and denote by H = Stab(x) its stabilizer in G. Denote respectively by g, h the Lie algebras of G and H. Let g = s r be a Levi decomposition of g, where s is semi-simple and r is the solvable radical of g. Denote by s nc the non-compact semi-simple factor of s, by s c the compact one and let n be the nilpotent radical of g. Note that n is an ideal of g. Let us denote respectively by S, S nc , S c , R and N the connected Lie sub-groups of G associated to s, s nc , s c , r and n.

Let a be a Cartan subalgebra of s associated with a Cartan involution Θ. Consider s = s 0 ⊕ α∈∆ s α = a ⊕ m ⊕ α∈∆ s α the root space decomposition of s, where ∆ is the set of roots of (s, a). Denote respectively by ∆ + , ∆ -the set of positive and negative roots of s for some chosen notion of positivity on a * . Then

s = s -⊕ a ⊕ m ⊕ s + , where s + = α∈∆ + s α and s -= α∈∆ -s α .
For every α ∈ a * , consider

g α = {X ∈ g, ∀H ∈ a : ad H (X) = α(H)X}.
We say that α is a weight if g α = 0. In this case g α is its associated weight space. As [g, r] ⊂ n (see [START_REF] Jacobson | Lie algebras[END_REF]Theorem 13]) then, for every α = 0, g α = s α ⊕ n α , where

n α = {X ∈ n, ∀H ∈ a : ad H (X) = α(H)X}.
Moreover, the commutativity of a together with the fact that finite dimensional representations of a semi-simple Lie algebra preserve the Jordan decomposition implies that elements of a are simultaneously diagonalisable in some basis of g.

Thus g = g 0 ⊕ α =0 g α .
Finally we will denote respectively by A, S + the connected Lie subgroups of G corresponding to a and s + .

2.2. General facts. We will prove some general results about the conformal group G. We start with the following general fact: Proposition 2.1. We have that [s, n] = [s, r]. In particular the sub-algebra s n is an ideal of g.

Proof.

For this, let us consider the semi-simple S-representation in GL(r). It preserves n and thus has a supplementary invariant subspace. But [g, r] ⊂ n so automorphisms of r act trivially on r/n and hence [s, g] ⊂ s ⊕ [s, n] ⊂ s n. We deduce that s n is an ideal of g.

Next we will prove:

Proposition 2.2. The non-compact semi-simple factor S nc of S is non trivial.

Let us first start with the following simple observation: Proposition 2.3. If a conformal diffeomorphism f of (M, g) preserves a volume form ω on M , then it preserves a metric in the conformal class of g.

Proof. Let f be a diffeomorphism preserving the conformal class [g] and a volume form ω on M . Denote by ω g the volume form defined on M by the metric g. On the one hand, there exists a C ∞ real function φ such that ω = e φ ω g . Hence ω is the volume form defined by the metric e Assume that the non-compact semi-simple factor S nc is trivial. Then by [START_REF] Zimmer | Ergodic theory and semisimple groups[END_REF]Corollary 4.1.7] the group G is amenable. So it preserves a regular Borel measure µ on the compact manifold M . It is in particular a quasi-invariant measure with associated rho-function ρ 1 = 1 (in the sense of [START_REF] Bekka | Kazhdan's property (T)[END_REF]). Let now ω g be the volume form corresponding to the metric g. As the group G acts conformally and the action is C ∞ , the measure ω g is also quasi-invariant with C ∞ rho-function ρ 2 (see [START_REF] Bekka | Kazhdan's property (T)[END_REF]Theorem B.1.4]). Again by [START_REF] Bekka | Kazhdan's property (T)[END_REF]Theorem B.1.4], the measures µ and ω g are equivalent and dµ dωg = 1 ρ2 . This shows that µ is a volume form. Then one use Corollary 2.4 to get the Proposition 2.2.

In the general case the essentiality of the action ensure the non discreetness of the stabilizer H.

Proposition 2.5. The stabilizer H is not discrete.

Proof. If it was not the case then H would be a uniform lattice in G. But as the action is essential, there is an element h ∈ H that does not preserve the metric on g/h. So |det (Ad h )| = 1 contradicting the unimodilarity of G.

To finish this part let us prove the two following important Lemmas that will be used later in the paper: Lemma 2.6. Let π : S nc -→ GL(V ) be a linear representation of S nc into a linear space V . Then, the compact orbits of S nc are trivial.

Proof. Assume that S nc has a compact orbit C ⊂ V . Then the convex envelope Conv(C ∪ -C) is an S nc -invariant compact convex symmetric set with non empty interior. Thus the action of S nc preserves the Minkowski gauge . (which is in fact a norm) of Conv(C ∪ -C). But Isom (Conv(C ∪ -C), . ) is compact. So the restriction of the representation π to Conv(C ∪ -C) gives rise to an homomorphism from a semi-simple group with no compact factor to a compact group and hence is trivial.

Lemma 2.7. A linear representation π : s nc -→ gl(V ) of s nc into a linear space V is completely determined by its restriction to a ⊕ m ⊕ s + . More precisely, π snc (V ) = Vect π a⊕m⊕s+ (V ) .

Proof. It is in fact sufficient to show that π s-(V ) ⊂ Vect π a⊕s+ (V ) . For that, fix x ∈ s -α ⊂ s -and let a ∈ a such that Rx ⊕ Ra ⊕ RΘ(x) ∼ = sl(2, R) (see for example [15, Proposition 6.52]). Thus the restriction of π to Rx ⊕ Ra ⊕ RΘ(x) is isomorphic to a linear representation of sl(2, R) into V . Using Weyl Theorem we can assume without loss of generality that this last is irreducible. But irreducible linear representations of sl(2, R) into V are unique up to isomorphism (see for instance [START_REF] Hall | Lie groups, Lie algebras, and representations[END_REF]Theorem 4.32]). It is then easy to check that they verify π(x)(V ) ⊂ Vect π Ra⊕RΘ(x) (V ) (see [START_REF] Hall | Lie groups, Lie algebras, and representations[END_REF]Examples 4.2]). This finishes the proof.

Lie algebra formulation

3.1. Enlargement of the isotropy group. As the manifold G/H is compact, the isotropy subgroup H is a uniform subgroup of G. If H was discrete then it is a uniform lattice and in this case G would be unimodular. In the non discrete case, this imposes strong restrictions on the group H. When H and G are both complex algebraic it is equivalent to being parabolic i.e contains maximal solvable connected subgroup of H. In the real case, Borel and Tits [START_REF] Borel | Groupes réductifs[END_REF] proved that an algebraic group H of a real linear algebraic group G is uniform if it contains a maximal connected triangular subgroup of G. Recall that a subgroup of G (respectively a sub-algebra of g) is said to be triangular if, in some real basis of g, its image under the adjoint representation is triangular.

Let H * = Ad -1 Ad(H) Zariski be the smallest algebraic Lie subgroup of G containing H. By [12, Corollary 5.1.1], the Lie algebra h * of H * contains a maximal triangular sub-algebra of g. The sub-algebra (a ⊕ s + ) n being triangular, we get the following fact: Fact 3.1. Up to conjugacy, the sub-algebra h * contains (a ⊕ s + ) n.

Consider the vector space Sym(g) of bilinear symmetric forms on g. The group G acts naturally on Sym(g) by g.Φ(X, Y ) = Φ(Ad g -1 X, Ad g -1 Y ). Let ., . be the bilinear symmetric form on g defined by

X, Y = g (X * (x), Y * (x)) ,
where g is the pseudo-Riemannian metric, X * , Y * are the fundamental vector fields associated to X and Y and x is the point fixed previously. It is a degenerate symmetric form with kernel equal to h.

Let P be the subgroup of G preserving the conformal class of ., . . It is an algebraic group containing H and normalizing the sub-algebra h. In particular, it contains H * : the smallest algebraic group containing H. Using Fact 3.1 we get that up to conjugacy, the Lie algebra p of P contains (a ⊕ s + ) n.

Proposition 3.2. The Cartan sub-group A does not preserve the metric ., . . Proof. First as h is an ideal of p then by taking quotient of both P and H by H

• , we can suppose that H is a uniform lattice of P and in particular that P is unimodular.

Assume that A preserves the metric ., . . On the one hand, the groups S + and N preserve the conformal class of ., . . On the other hand, they act on Sym(g) by unipotent elements. So the groups A, S + , and N preserve the metric ., . . But by Iwasawa decomposition (A S + ) is co-compact in S. Thus the S-orbit of ., . is compact in Sym(g) and hence trivial by Lemma 2.6. Therefore S and N are subgroups of P . This implies that for any p ∈ P , det (Ad p ) |g/p = 1. Indeed, the action of G on (s c + r)/n factors trough the product of the action of S c on s c by the trivial action on r/n. As P contains S and N , its action on g/p is a quotient of the action of S c on s c . But S c is compact, thus it preserves some positive definite scalar product and hence the determinant det (Ad p ) |g/p = 1. Now let h ∈ H such that Ad h does not preserve ., . . We have that

1 = det (Ad h ) |g/h = det (Ad h ) |g/p det (Ad h ) |p/h
Finally we get det (Ad h ) |p/h = 1 which contradicts the unimodularity of P .

3.2. Distortion. The group P preserves the conformal class of ., . . There exists thus an homomorphism δ : P → R such that: for every p ∈ P and every u, v ∈ g,

(1) Ad p (u), Ad p (v) = e δ(p) u, v = det (Ad p ) |g/h 2 n u, v
In particular if p ∈ P preserves the metric then δ(p) = 0 and

(2)

Ad p (u), Ad p (v) = u, v
Or equivalently

(3) ad p (u), v + u, ad p (v) = 0
It follows that if the action of p ∈ P on g is unipotent then δ(p) = 0. Therefore, the homomorphism δ is trivial on S -and N but not on A by Proposition 3.2. We continue to denote by δ the restriction of δ to A. We can see it alternatively as a linear form δ : a → R, called distortion, verifying: for every a ∈ a and every u, v ∈ g,

(4) ad a (u), v + u, ad a (v) = δ(a) u, v Definition 3.1.
Two weights spaces g α and g β are said to be paired if they are not ., . -orthogonal.

Definition 3.2. A weight α is a non-degenerate weight if g α is not contained in h.

Definition 3.3. We say that a subalgebra g is a modification of g if g projects surjectively on g/h. In this case g / (g ∩ h) = g/h.

Proposition 3.3. If the weight space g 0 is degenerate then up to modification, g is semi-simple and M = G/H is conformally flat.

Proof. On the one hand, g 0 ⊂ h implies that a ⊂ h. As h is an ideal of p, we get that s + = [s + , a] ⊂ h. On the other hand, r ⊂ g 0 + n ⊂ g 0 + [n, a] ⊂ h. Thus, up to modification, we can assume that g is semi-simple and that h contains a + s + . Now let α max be the highest positive root and let X ∈ g αmax . Then d 1 e X : g/h → g/h is trivial. Yet e X is not trivial. We conclude using [START_REF] Frances | Formes normales pour les champs conformes pseudoriemanniens[END_REF]Theorem 1.4].

A direct consequence of Equation 4, is that if g α and g β are paired then α+β = δ. This shows that if α is a non-degenerate weight then δ -α is also a non-degenerate weight. In particular if 0 is a non-degenerate weight, then g 0 and g δ are paired and hence δ is a non-degenerate weight. In fact: Proposition 3.4. If 0 is a non-degenerate weight then δ is a root. Moreover s δ ⊂ h.

Proof. First we will prove that the subalgebras a and n δ are ., . -orthogonal. Let a ∈ a such that δ(a) = 0. Using Equation 4 for a, u = a and v ∈ n δ , we get, a, ad a (v) = δ(a) a, v . But v preserves ., . , thus by Equation 3, δ(a) a, v = 0. Hence a, v = 0, for every v ∈ n δ . We conclude by continuity. Now if δ was not a root then s δ = 0 and g δ = n δ . Thus a and g δ are orthogonal. Which implies that a ⊂ h. But h is an ideal of p, so g δ = [g δ , a] ⊂ h. This contradicts the fact that g δ is paired with g 0 .

To finish we need to prove that s δ ⊂ h. If this was not the case then a would be orthogonal to g δ . Hence g δ ⊂ h which contradicts again the fact that g δ is paired with g 0 .

3.3. The isotropy group is big. From now and until the end we will suppose that the non-compact semi-simple part S nc of G is locally isomorphic to the Möbius group SO(1, n + 1). In this case the Cartan Lie algebra a is one dimensional and we have s nc = s -α ⊕ a ⊕ m ⊕ s α , where α is a positive root, a ∼ = R, m ∼ = so(n), and

s -α ∼ = s α ∼ = R n . Moreover, g ±α = s ±α ⊕ n ±α , g 0 = a ⊕ m ⊕ s c ⊕ r 0 , g β = n β for every β = 0, ±α and r = r 0 ⊕ β =0 n β .
In section 3.1 we saw that the isotropy group H is contained in the algebraic group P which turn out to be big i.e to contain the connected Lie groups A, S α and N . Our next result shows that the group H itself is big:

Proposition 3.5. The Lie algebra h contains a ⊕ s α ⊕ β =0 n β .
Proof. We have that a ⊂ h. Indeed, if 0 is a degenerate weight then we are done. If not, then δ is a root and a ⊂ g 0 is orthogonal to every g β with β = δ. From the proof of Proposition 3.4 we know that a and n δ are orthogonal. Thus it remains to show that a and s δ are orthogonal. For that, let x ∈ s δ then Θ(x) ∈ s -δ and [x, Θ(x)] = 0 in a. Now using Equation 3 and the fact that one of x or Θ(x) preserve ., . , we get ad x (Θ(x)), x = 0. But a is one dimensional so it is orthogonal to s δ .

To end this proof, we have that h is an ideal of p and so

a ⊕ s α ⊕ β =0 n β = a ⊕   a, s α ⊕ β =0 n β   ⊂ h ⊕ [h, p] ⊂ h.
As a consequence we get:

Corollary 3.6. If 0 is a non-degenerate weight, then δ = -α.

3.4.

A suitable modification of g. We will show that g admits a suitable modification g . This allows us to considerably simplify the proofs in the next section.

More precisely, we have:

Proposition 3.7. The solvable radical decomposes as a direct sum r = r 1 ⊕r 2 , where r 1 is a subalgebra commuting with the semi-simple factor s and r 2 is an s-invariant linear subspace contained in h. In particular g = s ⊕ r 1 is a modification of g.

To prove Proposition 3.7, we need the following lemma:

Lemma 3.8. We have [s, n] = [s, r] ⊂ h.

Proof of Lemma 3.8. First we prove that [n, g 0 ] ⊂ h. For this, note that by the Jacobi identity and the fact that n is an ideal of g, we have β =0 n β , g 0 = β =0 n β which in turn is a subset of h by Proposition 3.5. Thus one need to prove that [n 0 , g 0 ] ⊂ h. We know that n preserve the metric ., . . So using Equation 3 for p ∈ n 0 , u ∈ g 0 and v ∈ g δ gives us: ad p (u), v + u, ad p (v) = 0. But once again by Jacobi identity, the fact that n is an ideal of g and Proposition 3.5 we have ad p (v) ∈ g δ ∩ n = n δ ⊂ h. So ad p (u), v = 0, which means that [n 0 , g 0 ] is orthogonal to g δ . Using the fact that [n 0 , g 0 ] ⊂ g 0 and that g 0 is orthogonal to every g β 7

Date:

  May 29, 2023.

  2φ n g. On the other hand, we have f * e 2φ n g = e ψ e 2φ n g, for some C ∞ function ψ. Thus f * ω = e n 2 ψ ω. But, f preserves the volume form ω, so ψ = 0 which means that f preserves the metric e 2φ n g. As a consequence we get: Corollary 2.4. The conformal group G preserves no volume form on M .

for β = δ we get that [n 0 , g 0 ] ⊂ h.

Next we have that s c ⊂ g 0 thus [s c , n] ⊂ [g 0 , n] ⊂ h.

Finally we finish by proving that [s nc , n] ⊂ h. On the one hand we have,

On the other hand, as s nc is semi-simple we have by Lemma 2.7 that [s nc , n] ⊂ Vect ([a ⊕ m ⊕ s α , n]) ⊂ h.

Proof of Proposition 3.7. The subalgebra [s, n] = [s, r] is s-invariant, so it admits an s-invariant supplementary subspace r 1 in r. But s acts trivially on r/ [s, n] and thus it acts trivially on r 1 . We take r 1 to be the s-invariant subalgebra generated by r 1 (in fact the action of s on r 1 is trivial).

It is clear that r 1 is a direct sum of r 1 and r 1 : an s-invariant subspace of [s, n]. Consider r 2 to be the supplementary of r 1 in [s, n] = [s, r]. It is s-invariant and by Lemma 3.8 we have r 2 ⊂ h.

The Möbius conformal group: a classification theorem

This section is devoted to prove Theorem 1.1. We distinguish two situations: when m is contained in h and when it is not. In this last one, we first consider the case where only the non-compact semi-simple part S nc is non trivial. Then deduce from it the general case. From now and until the end we will assume, up to modification, that g = s ⊕ r 1 .

4.1. The Frances-Melnick case. We suppose that the sub-algebra m is contained in h. Then we have the following proposition: Proposition 4.1. M is conformally equivalent to the standard sphere S n or the Einstein universe Ein 1,1 .

Proof. Assume first that g 0 is contained in h. Then by Proposition 3.3, M is conformally flat and after modification, r = 0. Moreover, g/h ∼ = s -α . This is because

Now suppose that g 0 is not in h. In this case g -α = g δ is paired with g 0 . But a, m and n δ are contained in h so s -α is paired with s c ⊕ (r 0 ∩ r 1 ). Note that m acts on s -α ⊕ (s c ⊕ (r 0 ∩ r 1 )) by preserving the pairing (in fact the action of m preserves the metric ., . ). On the contrary for n ≥ 2, m ∼ = so(n) acts trivially on r 0 ∩ r 1 and transitively on s -α -{0}, so n = 1. As the metric is of type (p, q), we conclude that the projection of s c ⊕ (r 0 ∩ r 1 ) on g/h is ∼ = R. Thus, after modification g = so(1, 2) ⊕ R = u(1, 1), h = a ⊕ s α = R ⊕ R and hence M is, up to cover, conformally equivalent to Ein 1,1 .

4.2.

The non-compact semi-simple case. Here we suppose that m is not contained in h, the compact semi-simple part s c and the radical solvable part r 1 are both trivial. We will show: Proposition 4.2. The pseudo-Riemannian manifold M is conformally equivalent to Ein 3,3 By corollary 3.6, δ is a negative root. In particular δ = -α and g -α is paired with g 0 . In addition g = s -α ⊕ a ⊕ m ⊕ s α and a ⊕ s α ⊂ h. We have: Proposition 4.3. The root space s δ does not intersect h. In particular the metric is of type (n, n).

Proof. If it was the case then let 0 = X ∈ s δ ∩ h. We have [[X, s -δ ] , X] = s δ so s δ ⊂ h. This contradicts the fact that g δ is paired with g 0 .

Consider the bracket [., .] :

and .∨. : s α ×s -α -→ a its projections on m and a respectively. Direct computations give us: Lemma 4.4.

(

The Cartan involution identifies s α and s -α , which when identified with R n , m acts on them as so(n). In this case, the map .∨. can be seen as a bilinear symmetric map from R n × R n to R n , and when composed with α gives rise to an m-invariant scalar product ., . 0 on R n . Moreover, by Lemma 4.4, for every x, X ∈ R n , X ∧ x is the antisymmetric endomorphism of R n defined by X ∧x(y) = X, y 0 x-x, y 0 X.

Let x, X ∈ R n and consider P the plane generated by x, X. Then X ∧ x when seen as element of m ∼ = so(n) is the infinitesimal generator of a one parameter group acting trivially on the orthogonal P ⊥ of P with respect to the scalar product ., . 0 . Hence X ∧ x ∈ so(P ). More generally: Proposition 4.5. Let E be a linear subspace of R n and let x ∈ E. Consider c the Lie subalgebra of so(n) generated by {X ∧ x/X ∈ E}. Then c equals the Lie algebra linearly generated by {X ∧ X /X, X ∈ E}, which in turn equals so(E), the Lie algebra of orthogonal transformations preserving E and acting trivially on its orthogonal (with respect to ., . 0 ).

Proof. First we have c(E) ⊂ E and hence c ⊂ so(E). It is then sufficient to prove that c and so(E) have the same dimensions. For that let {x, X 2 , ..., X k } be a basis of E. Note that {X 2 ∧ x, ..., X k ∧ x, [X i ∧ x, X j ∧ x] , for 2 ≤ i < j ≤ k} are linearly independent. Thus c = so(E).

For every x = 0 ∈ s -α consider:

Proof. By Proposition 4.5 we have,

This implies that

Hence, the projections Θ(Z x )\{0}

x∈s-α form a partition of g/h.

Next we prove:

Proposition 4.7. The pseudo-Riemannian manifold M is conformally flat.

Proof. We need to prove that the Weyl tensor W (or the Cotton tensor C if the dimension of M is 3) vanishes. Actually we will just make use of their conformal invariance property. Namely: if f is a conformal transformation of M then, (5)

We denote by x the projection in g/h of an element x ∈ g. A direct application of Equation 5gives us:

(1) W(x, ȳ, z) = 0 for every x, y, z ∈ s -α ;

(2) W(x, ȳ, m) = 0 for every x, y ∈ s -α and every m ∈ m;

Then, from Equation 5we obtain:

In other words W(x, m 1 , m 2 ) ∈ Θ(Z x ). Now let x, y ∈ s -α , X ∈ s α and m ∈ m. Then again Equation 5gives us:

But W(x, [X, ȳ] , m) ∈ Θ(Z x ) and W([X, x] , ȳ, m) ∈ Θ(Z y ). Thus, Proposition 4.6 gives us:

(1) If y / ∈ Θ(Z x ) then W(x, [X, ȳ] , m) = 0; (2) In the case y ∈ Θ(Z x ) and X ∈ Z x , we have W(x, [X, ȳ] , m) = 0 (3) If y ∈ Θ(Z x ) and X / ∈ Z x . Then because Θ(X) / ∈ Θ(Z x ) we have:

So as a conclusion we get W = 0.

We finish this section by proving Proposition 4.2: Proof of Proposition 4.2. First note that if n = 1 then m = 0. Thus we assume n ≥ 2. So far we have seen that M = SO(1, n + 1)/H is a conformally flat pseudo-Riemannian manifold of signature (n, n). Since the Lie algebra h contains a + s α , the group H

• is cocompact in SO(1, n + 1). Therefore SO

• is connected and compact, with a connected isotropy and hence simply connected. As M is connected, it covers SO

• and thus equals it.

On the one hand, the Einstein universe Ein n,n is simply connected. Thus M is identified to Ein n,n . So SO(1, n + 1) acts transitively on Ein n,n with isotropy H. By Montgomery Theorem [19, Theorem A] any maximal compact subgroup in SO(1, n + 1), e.g. K 2 = SO(n + 1), acts transitively on S n × S n the two fold cover of Ein n,n .

On the other hand, the conformal group of Ein n,n is SO(n+1, n+1). A maximal compact subgroup of it is K 1 = SO(n + 1) × SO(n + 1). Up to conjugacy , we can assume K 2 ⊂ K 1 . Therefore, K 2 = SO(n + 1) acts via a homomorphism ρ = (ρ 1 , ρ 2 ) : SO(n + 1) → SO(n + 1) × SO(n + 1). If SO(n + 1) is simple, then:

-either ρ 1 or ρ 2 is trivial and the other one is bijective, in which case ρ(SO(n + 1)) does not act transitively on S n × S n , -or both are bijective, and ρ(SO(n + 1) is up to conjugacy in SO(n + 1) × SO(n + 1) the diagonal {(g, g)/g ∈ SO(n + 1)}. The latter, too, does not act transitively on S n × S n .

Hence SO(n + 1) must be non-simple which implies n = 1 or n = 3. but n = 1 was excluded, and then remains exactly the case n = 3, for which M is conformally equivalent to Ein 3,3 . 4.3. The general case. In this section we will show Theorem 1.1 in the general case. We suppose that g = s -α ⊕ a ⊕ m ⊕ s α ⊕ s c ⊕ r 1 . Let us denote by m 0 = m ∩ h so that so(1, n + 1) ∩ h = a ⊕ s α ⊕ m 0 . A priori the subalgebra m 0 could be of any dimension in m. Nevertheless the hypothesis m ⊂ h restricts drastically the possibilities. So we have: Proof. If n = 2 then m = so(2). Hence [p, s α ] = a ⊕ m for any non null p ∈ s -α . Recall that s -α preserves the metric so by applying Equation 3 for p = v ∈ s -α , u ∈ s α we get s -α , m = 0. Thus m ⊂ h which contradicts our hypothesis.

Assume that n ≥ 3 and suppose that m 0 has codimension less then n-1. Denote by M 0 the connected subgroup of SO(n) corresponding to m 0 .

If the action of M 0 on s -α ∼ = R n is reducible then M 0 preserves the splitting R d × R n-d and hence is contained in SO(d)×SO(n-d). Thus M 0 has codimension bigger than the codimension of SO(d) × SO(n -d) which in turn achieves its minimum if d = 1 or n -d = 1 and hence M 0 = SO(n -1). One can identify m 0 with so(E) for some n-1 dimensional linear subspace E of s -α . Let then e ∈ s -α such that s -α = Re ⊕ E. Fix a non zero element x ∈ Θ(E), we have ad x (e), X + e, ad x X = 0 for every X ∈ s -α and so in particular e, ad x e = 0. In addition by Proposition 4.5, [E, Θ(E)] = a ⊕ m 0 ⊂ h thus ad x e, X = 0 for every X ∈ E and hence ad x e is orthogonal to s -α . This implies that x ∧ e ∈ h ∩ m = m 0 = so(E) which contradicts the fact that x ∧ e is the infinitesimal rotation of the plane Re ⊕ Rx.

The last case to consider is when M 0 acts irreducibly. Let m ∈ m 0 , X ∈ s -α and y ∈ s c ⊕ r 1 then ad m (X), y + X, ad m y = 0. But ad m y = 0 and hence s c ⊕ r 1 is orthogonal to [m 0 , s -α ] which is equal to s -α by irreducibility. Thus s c ⊕ r 1 ⊂ h and we are in the non-compact semi-simple case. Therefore n = 3 and m ∼ = so(3). Non trivial Sub-algebras of so(3) have dimension one and are reducible. So the only left possibility is m 0 = m ∼ = so(3) which show that m ⊂ h and this is a contradiction.

End of Proof of Theorem 1.1. By Proposition 4.8, m 0 is of codimension n in m. But s -α is paired with g 0 = a ⊕ m ⊕ s c ⊕ (r 1 ∩ r 0 ). Thus s c ⊕ (r 1 ∩ r 0 ) ⊂ h and we are also in the non-compact semi-simple case. Therefore n = 3 and M is conformally equivalent to Ein 3,3 .

[] Mehdi Belraouti Faculté de Mathématiques,, USTHB, BP 32, El-Alia,, 16111 Bab-Ezzouar, Alger (Algeria)

E-mail address: mbelraouti@usthb.dz Mohamed Deffaf Faculté de Mathématiques,, USTHB, BP 32, El-Alia,, 16111 Bab-Ezzouar, Alger (Algeria)

E-mail address: mdeffaf@usthb.dz Yazid raffed Faculté de Mathématiques,, USTHB, BP 32, El-Alia,, 16111 Bab-Ezzouar, Alger (Algeria)

E-mail address: yazidsaid.raffed@usthb.edu.dz

Abdelghani Zeghib UMPA, ENS de Lyon, France E-mail address: abdelghani.zeghib@ens-lyon.fr


Fabio Mensi

Angelo Furno

Rémy Cazabet

Traffic speed prediction in the Lyon area using DCRNN

Keywords: Traffic prediction, Machine learning on graphs, Graph neural networks

2 Graph Neural Networks, the idea In many domains, data is naturally represented in the form of graphs. In chemistry, molecules are made of atoms linked via chemical bonds, in e-commerce, costumers and products are linked via their consumer relationship. Novel deep learning methods have been introduced to address those problems where data

Introduction

In this paper, we deal with the problem of traffic speed prediction, i.e., forecasting, based on relevant historical data (e.g., previous speed values, topological features, etc.) up to a certain time t, what will be the average speed on a given set of road segments at time t + T , where T is the prediction horizon.

More specifically, we focus on short term prediction (i.e., a maximum T of one hour) for a subset of road segments in the road network of Lyon, France, by exploiting the past history of observed traffic speeds from a real dataset.

Lyon road network

Our dataset consists in floating car data, reconstructed from GPS trajectories collected from 20,000 cars on an average working day. Data is heterogeneously distributed and most of the signal relates to a minor subset of road segments during daytime. To address this issue we have developed a filtering pipeline to select a subset of road segments that meet several criteria, including high data availability, non-stationarity and noise.

From the initial road network, consisting in 317,693 nodes, we extract a subgraph of 180 road segments, as highlighted in Fig. 1. This subset is interested by several congestion phenomena, making the traffic speed prediction task particularly challenging and the adoption of deep learning techniques, such as Graph Neural Networks (GNN), well justified. lies on graph by exploiting their underlying network structure [START_REF] Scarselli | The graph neural network model[END_REF]. These methods are generally referred as GNNs and they are typically implemented to solve problems such as node classification, graph classification or link prediction.

We focus our attention on a specific kind of GNN, namely Diffusion Convolutional Recurrent Neural Network (DCRNN), that has already shown promising results in traffic speed forecasting settings.

DCRNN

The DCRNN is a type of GNN specifically aimed at tackling spatiotemporal forecasting tasks, first introduced in an article by Y. Li et al. [START_REF] Li | Diffusion convolutional recurrent neural network: Data-driven traffic forecasting[END_REF]. In the original paper it was described and evaluated on a traffic speed prediction problem, reporting state-of-the-art performances.

Speed data can be represented as graph signals (time series) X t laying on the road network, modelled as a directed graph G, where road segments are represented as nodes and intersections as edges.

Diffusion Convolution operation

Spatial dependency is modelled following an analogy with a diffusion process.

Given a graph signal X ∈ R N ×P , with N being the number of nodes in the graph, and the diffusion transition matrices D -1 O W , D -1 I W T , with D O representing the out-degree diagonal matrix, D I the in-degree diagonal matrix, and W the adjacency matrix of the directed graph G, the convolution operation is defined as:

X :,p f θ = K-1 k=0 (θ k,1 (D -1 O W ) k + θ k,2 (D -1 I W T ) k )X :,p for p ∈ {1, ..., P } (1)
The filtered signal of a node i (1 ≤ i ≤ N ) is the result of combinations of the signals of the neighbors in the graph up to a distance of K steps from the node, weighted by the parameter θ ∈ R K×2 .

DCGRU cell

The diffusion convolution operation accounts for the spatial dynamics component of the problem and its implementation is directly connected with the temporal dynamics part.

The temporal dependency is modelled through a recurrent neural network variant, the gated recurrent unit (GRU) [START_REF] Bahdanau | Neural machine translation by jointly learning to align and translate[END_REF]. To combine spatial and temporal modelling each matrix multiplication operation is replaced by the the diffusion convolution operation, described in the previous equation. The resulting modified GRU cell, called DCGRU, can be defined by the following equations:

r (t) = σ(Θ r G [X (t) , H (t-1) ] + b r ) u (t) = σ(Θ u G [X (t) , H (t-1) ] + b u ) (2)
C (t) = tanh(Θ C G [X (t) , (r (t) H (t-1) )] + b c ) H (t) = u (t) H (t-1) + (1 -u (t) ) C (t)
where X (t) , H (t) are the input and output (or activation) at time t, r (t) , u (t) are the reset gate and update gate, and C (t) is the candidate output, which contributes to the new output based on the value of the update gate u (t) .

Results in the Lyon area

We have applied the DCRNN method to our case study, using data from October-November 2017, and compared it with simple baselines, such as historical average and naive forecasting, and other more traditional methods, such as ARIMA (Au-toRegressive Integrated Moving Average) and VAR (Vector AutoRegression).

In Table 1, the performances of these techniques are reported according to commonly used error metrics. Performance improvement can be observed when using DCRNN, especially with shorter horizons and the Mean Average Error (MAE) metric (the metric the neural network optimizes during training). An additional study has been performed on two subareas, where we compare DCRNN to other classical deep learning methods obtaining similar results, with DCRNN exhibiting better performance overall.

Reusable DCGRU

In order to allow future model developments, the DCGRU cell should be readily usable, independently from its original context. Starting from the original implementation [START_REF] Li | Diffusion convolutional recurrent neural network: Data-driven traffic forecasting[END_REF], we have built the DCGRU cell using the latest version of Tensorflow, a popular machine learning library, in such a way that is compatible with Keras, a deep learning API, and does not rely on deprecated libraries.

Our goal is to provide code that allows for quick model upgrading and can be integrated with minimal effort4 . We have tested the newly implemented DCGRU layer on the prediction of a graph-based synthetic signal and compared it with other commonly used deep learning architectures. The DCGRU layer obtained the best performance, showing potential for employment in similar tasks.

Future perspectives

We aim at upgrading the current DCGRU cell to make it able to exploit information other than traffic speed or graph-based signals. For example non-graph related temporal data (in a traffic prediction context this could mean weather, holidays, weekday, etc...), static graph-related features (e.g. road structure) or the adoption of attention mechanisms [START_REF] Veličković | Graph attention networks[END_REF] in the diffusion convolution operation.

Other efforts could be dedicated to redefine the traffic speed prediction problem in order to meaningfully consider those nodes that have been excluded in the filtering step (mostly urban road segments).

Fig. 1 .

 1 Fig. 1. Lyon subgraph, blue markers represent nodes

Table 1 .

 1 Performance comparison for different forecasting approaches, applied on the

	Lyon area subgraph
	T Metric Hist Avg. Naive ARIMA VAR DCRNN
	MAE	6.27	4.01	4.36	4.13	3.23
	15min MSE 152.83 79.83 85.84 69.33 59.80
	MAPE 9.70% 8.37% 9.63% 9.23% 7.39%
	MAE	6.27	5.43	5.78	5.05	4.07
	30min MSE 152.83 151.48 147.53 131.04 101.82
	MAPE 9.70% 12.11% 13.42% 12.07% 10.07%
	MAE	6.27	6.81	7.01	5.76	4.74
	60min MSE 152.83 295.93 256.92 232.44 167.13
	MAPE 9.70% 19.00% 19.35% 16.32% 14.54%

Code available at github.com/mensif/DCGRU Tensorflow2


João Pedro Baptista

Anabela Gradim

Romy Sauvayre
email: romy.sauvayre@uca.fr

Dissemination of a "Fake Miracle Cure" against COVID-19 on Twitter: The Case of Chlorine Dioxide

Keywords: diffusion, beliefs, chlorine dioxide, pseudoscience, science, complementary and alternative medicine, rationality, social network analysis, Twitter, misinformation, controversy

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction: Changing Beliefs: From Infodemic to "Miracle Cures"

Some beliefs persist and spread from century to century, while others fall into oblivion. The last two decades have revealed a growing increase in misinformation on the internet following the 1990s [START_REF] Barkun | Conspiracy theories as stigmatized knowledge[END_REF] and then on social media [START_REF] Zhou | Characterizing the dissemination of misinformation on social media in health emergencies: An empirical study based on COVID-19[END_REF]). Misinformation can be considered a virus that spreads out of control. According to the journalist John Zarocostas (2020), while the "infodemic" increases following each pandemic, that of COVID-19, which appeared in December 2019, has the particularity of being based on the preponderant influence of social networks. The COVID-19 pandemic then contributed to the dissemination of new beliefs and adherence to new cults, particularly in the field of health (MIVILUDES 2021). The content of these beliefs then evolves with our societies to include additional current aspects, such as distrust of authorities, conspiracy, or antivaccinism. Adherence to conspiratorial beliefs, alternative medicine, and rejection of political or medical authorities have affected the way citizens accept recommendations concerning COVID-19 (barrier gestures, physical distancing, etc.) and vaccination [START_REF] Soveri | Unwillingness to engage in behaviors that protect against COVID-19: The role of conspiracy beliefs, trust, and endorsement of complementary and alternative medicine[END_REF]. The misinformation widely disseminated via social networks amplified vaccine hesitancy [START_REF] Kanozia | Fake news", religion, and COVID-19 vaccine hesitancy in India, Pakistan, and Bangladesh[END_REF], which limited vaccine adherence (believing in the safety of vaccines) and acceptance (receiving vaccines).

At the same time, the craze for "miracle cures" grew during the COVID-19 pandemic. A "miracle cure" is defined in the present study as a type of complementary and alternative medicine (CAM) that offers unfailing effectiveness against a fatal disease with safety and Soc. Sci. 2023, 12, 320 2 of 13 without side effects, but the claimed efficiency is not proven by the science community. Beliefs and promotion are then linked. Some people promote a remedy as a "miracle cure", and some may believe in its "miracle" effectiveness.

In French-speaking countries, where the study takes place, the number of French newspaper articles extracted from Europresse, the requests by French internet users on Google extracted from Google Trends, and the French messages posted on Twitter extracted using the Twitter application programming interface (API) V2 peaked during the COVID-19 pandemic (Figure 1). Indeed, Europresse counted 9787 articles published between 2006 and 2021, mainly concentrated in 2020 (2319 articles) concomitant with the COVID-19 pandemic. The same evolution can be seen on Twitter: 87,564 tweets mentioning the terms "miracle cure" or "miracle cures" were posted between 2006 and 2021, and 52% were posted in 2020 and 2021 (i.e., 45,603 tweets).

At the same time, the craze for "miracle cures" grew during the COVID-19 pandemic. A "miracle cure" is defined in the present study as a type of complementary and alternative medicine (CAM) that offers unfailing effectiveness against a fatal disease with safety and without side effects, but the claimed efficiency is not proven by the science community. Beliefs and promotion are then linked. Some people promote a remedy as a "miracle cure," and some may believe in its "miracle" effectiveness.

In French-speaking countries, where the study takes place, the number of French newspaper articles extracted from Europresse, the requests by French internet users on Google extracted from Google Trends, and the French messages posted on Twitter extracted using the Twitter application programming interface (API) V2 peaked during the COVID-19 pandemic (Figure 1). Indeed, Europresse counted 9787 articles published between 2006 and 2021, mainly concentrated in 2020 (2319 articles) concomitant with the COVID-19 pandemic. The same evolution can be seen on Twitter: 87,564 tweets mentioning the terms "miracle cure" or "miracle cures" were posted between 2006 and 2021, and 52% were posted in 2020 and 2021 (i.e., 45,603 tweets). During the COVID-19 pandemic, promises of healing through "miracle cures" have multiplied and spread. As revealed by Google Fact Check Explorer (Fact Check Tools n.d.) on 14 December 2021, there are three types of "miracle cures." The first are based on medicinal substances, such as molnupiravir, budesonide, bromhexine, ivermectin, and hydroxychloroquine [START_REF] Baker | From COVID-19 Treatment to Miracle Cure: The Role of Influencers and Public Figures in Amplifying the Hydroxychloroquine and Ivermectin Conspiracy Theories during the Pandemic[END_REF]. The second type of "cures" is chemical substances promoted long before the pandemic, such as hydrogen peroxide promoted by Dr. William [START_REF] Douglass | Hydrogen Peroxide-Medical Miracle[END_REF] or chlorine dioxide promoted by Jim V. Humble. The third type of "cure" is natural, proposing the ingestion of red onions, condensed milk, honey, pepper, ginger, vegetable soup, tea, hot water with lemon, and many other foods. These natural "cures" also invite the inhalation of salt water, clove, or orange vapor.

It might seem surprising to see rational individuals adhering to these types of cure not proven by science. However, this tendency to consider alternative promises of healing is found among a majority of French people. Indeed, in 2019, 68% of French people believed in the benefits of CAM, such as osteopathy, acupuncture, homeopathy, hypnosis, phytotherapy, sophrology, and meditation [START_REF] Odoxa | Baromètre santé 360. Les médecines alternatives et complémentaires[END_REF]. Almost half of the French (44%) even consider them to be more effective than conventional medicine. This leads some to replace conventional medicine with alternative medicine. Parallel to this trend observed During the COVID-19 pandemic, promises of healing through "miracle cures" have multiplied and spread. As revealed by Google Fact Check Explorer (Fact Check Tools n.d.) on 14 December 2021, there are three types of "miracle cures". The first are based on medicinal substances, such as molnupiravir, budesonide, bromhexine, ivermectin, and hydroxychloroquine [START_REF] Baker | From COVID-19 Treatment to Miracle Cure: The Role of Influencers and Public Figures in Amplifying the Hydroxychloroquine and Ivermectin Conspiracy Theories during the Pandemic[END_REF]. The second type of "cures" is chemical substances promoted long before the pandemic, such as hydrogen peroxide promoted by Dr. William Campbell [START_REF] Douglass | Hydrogen Peroxide-Medical Miracle[END_REF] or chlorine dioxide promoted by Jim V. Humble. The third type of "cure" is natural, proposing the ingestion of red onions, condensed milk, honey, pepper, ginger, vegetable soup, tea, hot water with lemon, and many other foods. These natural "cures" also invite the inhalation of salt water, clove, or orange vapor.

It might seem surprising to see rational individuals adhering to these types of cure not proven by science. However, this tendency to consider alternative promises of healing is found among a majority of French people. Indeed, in 2019, 68% of French people believed in the benefits of CAM, such as osteopathy, acupuncture, homeopathy, hypnosis, phytotherapy, sophrology, and meditation [START_REF] Odoxa | Baromètre santé 360. Les médecines alternatives et complémentaires[END_REF]. Almost half of the French (44%) even consider them to be more effective than conventional medicine. This leads some to replace conventional medicine with alternative medicine. Parallel to this trend observed among citizens, researchers are increasingly interested in the beneficial effects of CAM. A systematic review [START_REF] Badakhsh | Complementary and alternative medicine therapies and COVID-19: A systematic review[END_REF] published in September 2021 showed that CAM improved the physical and psychological symptoms of patients with COVID-19. How then can these "miracle cures" be considered as beliefs when science demonstrates the effectiveness of some of them?

Objective

The objective of this study is to explore the boundary between science and belief based on the case of chlorine dioxide, a toxic bleaching agent for textiles or paper that also has disinfectant properties (water, surfaces) presented as a "miracle cure" particularly effective against COVID-19. The aim is to measure the spread of misinformation and fact-checking of a "miracle cure" presented as scientifically proven to the French-speaking people using Twitter.

First, the two primary forms of chlorine dioxide promoted will be described: the Miracle Mineral Solution (MMS) and Chlorine Dioxide Solution (CDS). Second, the content of messages posted in French on Twitter about chlorine dioxide and the links between users (tweeters) will be analyzed.

Using these data and academic articles published on chlorine dioxide, this study aims to explore how a belief perceived as irrational (ingesting a bleach derivative to treat COVID-19) can be legitimized by a scientific discourse and what consequences this has in the dissemination of "miracle cures" on a social network such as Twitter.

While the dissemination of fake miracle cures on social media can generate public health problems, to the best of our knowledge, there is a lack of studies on this issue. While health misinformation has been explored [START_REF] Waszak | The spread of medical fake news in social media-The pilot quantitative study[END_REF][START_REF] Rovetta | Global Infodemiology of COVID-19: Analysis of Google Web Searches and Instagram Hashtags[END_REF][START_REF] Suarez-Lledo | Prevalence of Health Misinformation on Social Media: Systematic Review[END_REF], the dissemination of fake miracle cure information has yet to be investigated. Thus, this study will reduce the research gap on this topic and provide insight into the mechanisms of the spread of medical beliefs and its interaction with science and misinformation.

Two Types of Chlorine Dioxide Miracle Cures

The Miracle Mineral Solution

The first type of "miracle cure" based on chlorine dioxide was "discovered" by Jim V. Humble, the founder of the Genesis II Church of Health and Healing (Genesis II Church of Health and Healing n.d.). Founded in 2010, this nonreligious church is located in Florida (United States v. Genesis II Church of Health and Healing 2020). "Bishop" Humble presented himself as a former aerospace "research engineer" (Humble and Loyd 2016) converted to "gold mining" [START_REF] Humble | MMS Health Recovery Guidebook[END_REF]. He promoted a miracle cure derived from bleach, which he called the Miracle Mineral Supplement (MMS), comprising chlorine dioxide (MMS1) or calcium hypochlorite (MMS2) [START_REF] Humble | MMS Health Recovery Guidebook[END_REF].

Humble said he discovered, as early as 1996, that chlorine dioxide quickly eradicated malaria during an expedition to South America. This product has, according to him, proven its effectiveness on "a wide range of diseases, including cancer, diabetes, hepatitis A, B, C, Lyme disease, MRSA, multiple sclerosis, Parkinson's, Alzheimer's, HIV/AIDS, malaria, autism, infections of all kinds, arthritis, acid reflux, kidney or liver disease, aches and pains, allergies, urinary tract infections, digestive problems, high blood pressure, obesity, parasites, tumors and cysts, depression, sinus problems, eye disease, ear infections, dengue fever, skin problems, dental issues, problems with prostate (high PSA), erectile dysfunction, and many others" [START_REF] Humble | MMS Health Recovery Guidebook[END_REF].

The Humble website also contains many testimonials classified by type of illness or condition. The most filled category is that of cancer with 111 testimonials (What Is MMS n.d.). That of the coronavirus contains 23 testimonials posted between March and November 2020. One attests to the effectiveness of the MMS treatment on COVID-19 in four doses of 24 drops (Coronavirus n.d.).

To convince his followers, as shown in the following quotation, Humble uses pseudoscientific arguments based on valid scientific knowledge but grants the MMS a broader scope of application than what science has demonstrated: "A great deal of evidence given by the FDA, EPA and various industrial corporations prove scientifically that MMS1 (chlorine dioxide) kills and or oxidizes pathogens and poisons in food, public water systems, hospitals, and even slaughter houses. It is our belief that the same thing can and does happen in the human body". [START_REF] Humble | MMS Health Recovery Guidebook[END_REF] In short, according to this fallacious reasoning, if chlorine dioxide purifies and disinfects water and surfaces, that it is used in sectors of activity (hospitals, laboratories, treatment of drinking water), then it can also disinfect the water contained in the body of anyone who ingests it.

However, as early as 2010, the French Agency for the Safety of Medicines and Health Products (ANSM) warned against the ingestion of chlorine dioxide, stating that "no medical efficacy of this product has been proven" (ANSM 2010). The Food and Drug Administration (FDA) did the same in 2019 (FDA 2019), specifying the risks of ingesting MMS that the agency had identified in consumers: "severe vomiting, severe diarrhea, life-threatening low blood pressure caused by dehydration, and acute liver failure after drinking these products" (FDA 2019). In 2020, the FDA fined Humble, Mark Grenon, and his sons (Joseph and Jordan) for the unauthorized sale of products for treating COVID-19 (Center for Drug Evaluation and Research 2020). The Grenons were responsible for selling and promoting MMS within the Genesis II Church before being arrested and charged on the basis of the FDA report (The United States Attorney's Office, Southern District of Florida 2021). Finally, a study published in March 2022 demonstrated the toxicity of this product [START_REF] Peltzer | Risk of chlorine dioxide as emerging contaminant during SARS-CoV-2 pandemic: Enzyme, cardiac, and behavior effects on amphibian tadpoles[END_REF].

The Chlorine Dioxide Solution

The second type of "miracle cure" based on chlorine dioxide is the Chlorine Dioxide Solution (CDS) "invented" by Andreas L. Kalcker. Kalcker presents himself as a "biophysicist" (Kalcker 2021) who has been studying chlorine dioxide for over a decade [START_REF] Kalcker | Bye Bye Covid[END_REF].

Kalcker tried to scientificize the promotion of this "miracle cure" by removing the spiritual dimensions brought by Humble and his church to give it medical and scientific probity. Indeed, in 2021, Kalcker published the book Bye Bye Covid (Kalcker 2021), as he said, which will change the "perspective on medicine" of readers. Chlorine dioxide is presented as a cure for COVID-19. Moreover, Kalcker claimed that the "CDS has brought healing in countless well-documented cases, and its effectiveness is now, despite what anyone might say, irrefutable" [START_REF] Kalcker | Bye Bye Covid[END_REF]). Kalcker compares the CDS and MMS. He claims that, unlike the MMS, the CDS does not cause adverse effects and that laboratory experiments with mice show that it "prolonged their lifespans up to 30%" [START_REF] Kalcker | Bye Bye Covid[END_REF]).

To reinforce this scientific "face" [START_REF] Goffman | On Face-Work: An Analysis of Ritual Elements in Social Interaction[END_REF], he founded the Coalición Mundial Salud y Vida (COMUSAV, "World Health and Life Coalition" in English) (Accueil n.d.), bringing together several thousand doctors using CDS (Insignares-Carrione et al. 2020). The COMUSAV has more than 46,000 subscribers to encrypted Telegram messaging. Members of the COMUSAV publish scientific articles and reports and give lectures alongside Kalcker and with members of the Liechtenstein Association for Science and Health (LVWG) (Home n.d.) based in Switzerland. The LVWG presents itself as an association aiming to bring together researchers, physicians, and funders. Its website provides numerous documents and publications oriented toward a single subject: demonstrating the effectiveness of chlorine dioxide for COVID-19. The scientific articles highlighted on the LVWG website and through speeches on the internet are signed by COMUSAV members attached to the LVWG or to the Jurica Medical Center (Centro Médico Jurica) located in Mexico. Dr. Aparicio-Alonso practices at this medical center. He specializes in traumatology and orthopedics (Conoce al Dr. Manuel Aparicio Alonso n.d.), and has published three peer-reviewed articles demonstrating the effectiveness of CDS for COVID-19 (Aparicio-Alonso et al. 2021a, 2021b, 2021c). However, the evidence for the effectiveness of chlorine dioxide is contested in other scientific works [START_REF] Baracaldo-Santamaría | Drug safety of frequently used drugs and substances for self-medication in COVID-19[END_REF][START_REF] De | Use of Ivermectin and Chlorine Dioxide for COVID-19 Treatment and Prophylaxis in Peru: A Narrative Review[END_REF].

Materials and Methods

To measure the diffusion of chlorine dioxide as a "miracle cure" against COVID-19, the terms "chlorine dioxide" and "COVID" were searched in French-speaking messages posted on Twitter between 1 December 2019 and 30 November 2021 and extracted using the Twitter API V2 Academic Research with a Python script request ("dioxide de chlore" AND COVID lang: fr). The following data were downloaded: tweet content, tweet ID, author ID, and creation date. A total of 1,252 messages were collected containing 596 unique tweets. An analysis of tweets per user was conducted to identify bot accounts, and fake users were removed from the collected data.

As "retweeting rapidly disseminates messages across user networks, creating a wider sphere of influence" [START_REF] Harrigan | Identifying influencers on social media[END_REF], several data were collected and aggregated to create a new indicator, a "diffusion metric", to measure the dissemination of messages. The larger the "diffusion metric" is, the more the message is spread on social media, and the more influential it is likely to be. To obtain a diffusion metric for each tweet, the number of retweets, mentions, likes, and replies was summed. This new metric was then used to select the 100 most influential tweets (excluding duplicates and tweets posted by robots), i.e., the 100 most retweeted, liked, mentioned, and replied-to tweets. Indeed, as many messages are relayed, focusing on influential tweets is more informative for the study of information dissemination.

Following this step, the influential tweets were labeled in two categories: (1) a sentiment analysis category (pro, against, or neutral) and ( 2) an abductive categorization according to the recurrence of the content and the main arguments presented (effectiveness of the treatment, scientific evidence, fight against misinformation, etc.). The abductive approach, inspired by the philosopher Charles S. Peirce [START_REF] Peirce | Collected Papers of Charles Sanders Peirce[END_REF], consists of grouping the content of messages as faithfully as possible by abstraction. This approach then makes it possible to avoid defining categories a priori and thus avoid being far from the observable. Finally, this has the advantage of limiting the biases likely to influence the classification process.

The categories of these 100 influential tweets were then applied to the retweets to build a file of edges and nodes to visualize the links between tweeters using Gephi 0.9.2 software. The most relevant indicator for this study is the eigenvector centrality [START_REF] Bonacich | Eigenvector-like measures of centrality for asymmetric relations[END_REF] because it makes it possible to highlight the links in a graph between the most influential tweeters and thus better visualize the dissemination of information in the social network.

Results

The 100 most influential tweets posted by 70 tweeters include 6287 "diffusion metrics" (likes, retweets, quotes, or replies). They are followed by a total of 1,620,808 people (followers). It should be noted that only 3 tweeters have an identity verified by Twitter and practice journalism. They alone have 1,306,259 followers. These influential tweets were posted between 24 April 2020 and 28 November 2021, with the highest activity in August, September, and November 2021 (Figure 2). Note that, in this influential sample of tweets, the higher the number of followers, following, participation in a discussion, the higher the diffusion metrics as Spearman's correlation analysis shows (respectively, r s = 0.488, p < 0.001; r s = 0.271 p < 0.05; r s = 0.502 p < 0.001).

As Table 1 shows, while the majority of tweets (77%) do not mention anyone in particular, almost a quarter of tweets (23%) mention people belonging to the COMUSAV promoting CDS (including Manuel Aparicio-Alonso and Andreas Kalcker). Mark Grenon, one of the promoters of the MMS, was also found. As Table 1 shows, while the majority of tweets (77%) do not mention anyone in particular, almost a quarter of tweets (23%) mention people belonging to the COMUSAV promoting CDS (including Manuel Aparicio-Alonso and Andreas Kalcker). Mark Grenon, one of the promoters of the MMS, was also found. A sentiment analysis (Figure 3) shows that these influential tweets mainly relay content favorable to chlorine dioxide ("pro chlorine dioxide") and, more rarely, comments coming in opposition to this product ("con chlorine dioxide"). The "pro chlorine dioxide" messages not only are more numerous but also include 3.2 times as many "diffusion metrics" than "con chlorine dioxide." In addition, the "pro" tweets have 9.2 times fewer followers than the "con" tweets but were nevertheless more widely relayed.

As shown in Table 2, influential tweets mostly promote the effectiveness of chlorine dioxide against COVID-19 (50%) or argue that scientific evidence demonstrates its effectiveness (15%). A total of 69% of tweets present chlorine dioxide as an effective treatment, administered by doctors or having been scientifically proven. A third type of tweet attempts to deny or contest the effectiveness of this treatment. These debunking tweets are the most numerous in the full data but are poorly represented (14%) among the most influential tweets. A sentiment analysis (Figure 3) shows that these influential tweets mainly relay content favorable to chlorine dioxide ("pro chlorine dioxide") and, more rarely, comments coming in opposition to this product ("con chlorine dioxide"). The "pro chlorine dioxide" messages not only are more numerous but also include 3.2 times as many "diffusion metrics" than "con chlorine dioxide". In addition, the "pro" tweets have 9.2 times fewer followers than the "con" tweets but were nevertheless more widely relayed. shown in Table 2, influential tweets mostly promote the effectiveness of chlorine dioxide against COVID-19 (50%) or argue that scientific evidence demonstrates its effectiveness (15%). A total of 69% of tweets present chlorine dioxide as an effective treatment, administered by doctors or having been scientifically proven. A third type of tweet attempts to deny or contest the effectiveness of this treatment. These debunking tweets are the most numerous in the full data but are poorly represented (14%) among the most influential tweets. Considering these data, fact-checking or any message aimed at reducing the spread of misinformation has a limited scope in the process of disseminating information on Twitter. This is confirmed by a network analysis of all the tweets and retweets collected. Indeed, the influence of tweets disseminated by a French media dedicated to fact-checking ("Fact and Furious", which disappeared at the end of 2022 (La Rédaction 2022)) had a limited and circumscribed scope (see green group in Figure 4). In contrast, the "pro chlorine dioxide" network has more links and more people relaying tweets with arguments in favor of chlorine dioxide (in purple in Figure 4). Arguments based on scientific evidence (claiming that a study demonstrates the effectiveness of the product) and doctors claiming the effectiveness of chlorine dioxide have a wider reach on this social network. The pro chlorine dioxide group, formed by relaying the same types of tweets, is carried by three main influencers. Few discussions take place with the fact-checking group of tweeters "con chlorine dioxide" (in green in Figure 4).

Therefore, the "pro chlorine dioxide" constitute a homogeneous group in terms of beliefs that relay more what they present as excellent news, namely, to have scientific proof of the effectiveness of the treatment, which also allows them to oppose anti-COVID vaccination. In addition, their followers are invited to click on the links pointing to the Odysee website dedicated to hosting videos similar to YouTube (Odysee Help 2021; Pezet 2020). Created in September 2020, Odysee had more than one million subscribers at the end of 2021. It notably hosts the very controversial conspiratorial documentary Hold-Up on COVID-19. Links posted on Twitter lead tweeters to Odysee, where they can see videos promoting the CDS treatment against COVID-19 by Dr. Aparicio-Alonso introducing himself as a doctor (13,000 views as of November 2021) and by Kalcker introducing himself as a researcher (49,000 views as of August 2021).

himself as a doctor (13,000 views as of November 2021) and by Kalcker introducing himself as a researcher (49,000 views as of August 2021).

Discussion

Dynamic of Misinformation Dissemination

This study aimed to explore the dissemination of false health information and, in particular, the promotion on Twitter of a "miracle cure" for COVID-19 during the pandemic. The number of tweets mentioning chlorine dioxide and COVID-19 had several peaks: a first peak in July 2020 and then three peaks in August, September, and November 2021. By comparison, a study looking at the evolution of Google queries about chlorine dioxide among the people of Mexico also showed peaks in July 2020 but differed at the level of the January 2021 peak [START_REF] Chejfec-Ciociano | Misinformation About and Interest in Chlorine Dioxide During the COVID-19 Pandemic in Mexico Identified Using Google Trends Data: Infodemiology Study[END_REF].

Furthermore, this study has shown by means of a network graph that messages promoting misinformation, even if they are likely to be quantitatively less numerous, spread more widely than those based on more reliable information. This finding aligns with previous results showing that social media promotes the widespread and rapid dissemination of conspiracy theories [START_REF] Cinelli | The COVID-19 social media infodemic[END_REF]) and false news [START_REF] Vosoughi | The spread of true and false news online[END_REF]) by their supporters [START_REF] Featherstone | Relationship of people's sources of health information and political ideology with acceptance of conspiratorial beliefs about vaccines[END_REF] and that misinformation about COVID-19 spreads more on social media if the content is presented positively [START_REF] Zhou | Characterizing the dissemination of misinformation on social media in health emergencies: An empirical study based on COVID-19[END_REF]). This phenomenon is also found in the dissemination of antivaccinism on the internet [START_REF] Kata | A postmodern Pandora's box: Anti-vaccination misinformation on the Internet[END_REF]. Finally, this observation was established on social networks during the pandemic: even if the antivaccine network is quantitatively smaller than the provaccine network, it extends more widely and thus tends to spread to a growing number of people, especially undecided people [START_REF] Johnson | The online competition between pro-and anti-vaccination views[END_REF]). In addition, the fact-checkers had few interactions with misinformation spreaders in the network; consequently, they had little influence on

Discussion

Dynamic of Misinformation Dissemination

This study aimed to explore the dissemination of false health information and, in particular, the promotion on Twitter of a "miracle cure" for COVID-19 during the pandemic. The number of tweets mentioning chlorine dioxide and COVID-19 had several peaks: a first peak in July 2020 and then three peaks in August, September, and November 2021. By comparison, a study looking at the evolution of Google queries about chlorine dioxide among the people of Mexico also showed peaks in July 2020 but differed at the level of the January 2021 peak [START_REF] Chejfec-Ciociano | Misinformation About and Interest in Chlorine Dioxide During the COVID-19 Pandemic in Mexico Identified Using Google Trends Data: Infodemiology Study[END_REF].

Furthermore, this study has shown by means of a network graph that messages promoting misinformation, even if they are likely to be quantitatively less numerous, spread more widely than those based on more reliable information. This finding aligns with previous results showing that social media promotes the widespread and rapid dissemination of conspiracy theories [START_REF] Cinelli | The COVID-19 social media infodemic[END_REF]) and false news [START_REF] Vosoughi | The spread of true and false news online[END_REF]) by their supporters [START_REF] Featherstone | Relationship of people's sources of health information and political ideology with acceptance of conspiratorial beliefs about vaccines[END_REF] and that misinformation about COVID-19 spreads more on social media if the content is presented positively [START_REF] Zhou | Characterizing the dissemination of misinformation on social media in health emergencies: An empirical study based on COVID-19[END_REF]. This phenomenon is also found in the dissemination of antivaccinism on the internet [START_REF] Kata | A postmodern Pandora's box: Anti-vaccination misinformation on the Internet[END_REF]. Finally, this observation was established on social networks during the pandemic: even if the antivaccine network is quantitatively smaller than the provaccine network, it extends more widely and thus tends to spread to a growing number of people, especially undecided people [START_REF] Johnson | The online competition between pro-and anti-vaccination views[END_REF]). In addition, the fact-checkers had few interactions with misinformation spreaders in the network; consequently, they had little influence on the COVID-19 misinformation spreading, consistent with previous findings [START_REF] Shahi | An exploratory study of COVID-19 misinformation on Twitter[END_REF].

The Dynamic of Adherence to Misinformation

can we explain the adherence to and diffusion of "fake miracle cures"? The ethics of belief consists, as expressed by the philosopher William K. Clifford [START_REF] Clifford | The Ethics of Belief[END_REF], of giving up believing without sufficient proof to do so. However, people adhering to or relaying pro chlorine dioxide messages can justify their adherence by means of assertions by experts (doctors, researchers) and academic journals. Consequently, they have factual elements whose epistemic value increases with the status of the people promoting this treatment (doctor or researcher) and the process by which the administration of the proof was carried out (scientific method with publication of conclusive results in an academic journal). The justification of their beliefs is therefore both rational (2003) and deontological.

In the case of chlorine dioxide dissemination, three elements justify the belief adherence of the followers. First, physicians such as Dr. Aparicio-Alonso present an unambiguous discourse on the efficacy of the product against COVID-19, as when he was interviewed by Stew Peters on his podcast [START_REF] Stew | À ÉCOUTER POUR TELLEMENT DE RAISONS!!!![END_REF].

Second, in the most retweeted chlorine dioxide video since the start of the pandemic dated 20 August 2021 (Astrid Stuckelberger-Andreas Kalcker-dioxyde de chlore 2021), Kalcker presents himself as a biophysicist who has been researching this product for 14 years. He claims to have found "the best oxygen provider in the blood that eliminates pathogens, especially viruses". He also claims that nearly 5000 physicians practicing in more than 25 countries are using the CDS with great success. He then recalls that they have proven the effects of this product without any toxicity or side effects by publishing, in particular, three peer-reviewed articles. One of these articles is based on a study conducted on a very large corpus of patients (100,000) in collaboration with many doctors practicing at hospitals. Kalcker said that the CDS had a preventive effect on 90,000 patients and their families, who then did not contract COVID-19. Like Aparicio-Alonso, he claims a recovery rate of 99.3%. Finally, he claims to have been his first human research subject since he saved his own "life more than once with this solution".

Third, internet users have "scientific evidence" published in peer-reviewed articles. They can be found on the Kalcker site but also on other journals' websites. Aparicio-Alonso is a coauthor of these three articles published in August and September 2021 in the journals Journal of Infectious Diseases and Therapy and International Journal of Multidisciplinary Research and Analysis. These journals, like thousands of others, are referenced neither by the Journal Citation Reports (JCR) based on the Web of Science (WoS) nor in the Scimago Journal & Country Rank (SJR) based on Scopus. Therefore, these journals do not benefit from a reputation in the scientific field. However, they are also not listed among the predatory journals ((New) List of Predatory Journals-2023 2020; Potential Predatory Scholarly Open-Access Journals 2021), which would fully disqualify their contribution.

Fake Miracle Cure or Scientific Discovery of an Effective Treatment?

During the COVID-19 pandemic, there have been numerous potential treatments (remdesivir, lopinavir, interferon, dexamethasone, tocilizumab, etc.), which have led to several randomized clinical trials (Solidarity, Discovery, Recovery, and others), which are the most esteemed method in the medical field. In 2020, Professor Didier Raoult's team published a study on hydroxychloroquine [START_REF] Gautret | Hydroxychloroquine and azithromycin as a treatment of COVID-19: Results of an open-label non-randomized clinical trial[END_REF], which led many scientists to believe that it was an effective remedy for COVID-19 and then to conduct clinical studies to test this hypothesis. In "normal science" [START_REF] Kuhn | The Structure of Scientific Revolutions[END_REF], sometimes a study transforms knowledge, and then a process of verification and controversy ensues to confirm or dispute the results obtained. In the case of hydroxychloroquine, posterity and replication studies have not confirmed the substance's effectiveness against COVID-19.

The only clinical trial on chlorine dioxide to our knowledge was registered on the National Institutes of Health (NIH) website in April 2020 (Insignares-Carrione and Bolano 2020) and led to the publication of an article in March 2021 in the unclassified Journal of Molecular and Genetic Medicine (Insignares-Carrione et al. 2021). The article demonstrates, based on a sample of 40 patients, the effectiveness of the substance against COVID-19. However, several indicators call for caution: the time submission and publication of the article was only 7 days, whereas the process requiring evaluation by experts is generally much longer (several months); the journal is not classified; and the authors are not affiliated with universities or research institutes.

On 17 November 2021, another article appeared in the esteemed journal BMC Public Health demonstrating the effectiveness of chlorine dioxide in the prevention and treatment of COVID-19 in a sample of 3630 Peruvians. Articles in favor of the effectiveness of chlorine dioxide subsequently accumulated. On 16 July 2022, an article published in the journal Oral Diseases [START_REF] Soriano-Moreno | Factors associated with the consumption of chlorine dioxide to prevent and treat COVID-19 in the Peruvian population: A cross-sectional study[END_REF], classified by the JCR and SJR, concluded that chlorine dioxide can be effective against SARS-CoV-2. On 9 December 2022, the JCR-and SJRranked journal BioScience Trends published an editorial [START_REF] Asakawa | Focusing on development of novel sampling approaches and alternative therapies for COVID-19: Are they still useful in an era after the pandemic?[END_REF]) and correspondence [START_REF] Cao | Can nasal irrigation with chlorine dioxide be considered as a potential alternative therapy for respiratory infectious diseases? The example of COVID-19[END_REF]) citing nasal chlorine dioxide as a possible treatment for COVID-19. The authors cite two of the three articles by Dr. Aparicio. The correspondence [START_REF] Cao | Can nasal irrigation with chlorine dioxide be considered as a potential alternative therapy for respiratory infectious diseases? The example of COVID-19[END_REF]) concludes that chlorine dioxide may be an effective remedy but that further research is needed.

In the case of chlorine dioxide, the research conducted offers contradictory results, and some researchers speak of controversy [START_REF] Liester | The chlorine dioxide controversy: A deadly poison or a cure for COVID-19[END_REF]. Then, chlorine dioxide could no longer be presented as a "fake miracle cure" but as a potentially therapeutic substance analyzed, discussed, and published as any other substance studied during the pandemic for treatment purposes. This raises important questions about the boundary between science and misinformation. Indeed, science makes it possible to draw this border. However, when scientists have unverified beliefs and try to publish them, the beliefs become potential knowledge until the process of replication allows the scientific community to decide. Until then, it has been difficult to say that this is misinformation. This leads unverified knowledge to spread widely on social networks, as this study shows.

Conclusions

The COVID-19 pandemic and the uncertainties regarding the disease have generated high expectations in terms of care. Researchers around the world have worked hard and published more than 50,000 articles on COVID-19 in 2020 and 2021 according to the WoS. Along with this effort by scientists, pseudoscience has offered "miracle cures" against COVID-19.

This study focuses on the examination of the dissemination a "miracle cure" on Twitter, namely, chlorine dioxide, a bleach derivative considered toxic by the FDA (FDA 2019) and in academic articles [START_REF] Peltzer | Risk of chlorine dioxide as emerging contaminant during SARS-CoV-2 pandemic: Enzyme, cardiac, and behavior effects on amphibian tadpoles[END_REF]. Chlorine dioxide has been promoted by a nonreligious movement since 1996 as a cure for all fatal diseases and recently by physicians as a cure for COVID-19 despite insufficient evidence of its efficacy against this disease [START_REF] Baracaldo-Santamaría | Drug safety of frequently used drugs and substances for self-medication in COVID-19[END_REF].

The results show that messages promoting misinformation, even if they are likely to be quantitatively less numerous, spread more widely than those based on more reliable information. This result aligns with those found in previous studies conducted on misinformation spreading. In addition, the analysis of messages posted on Twitter has shown that the dissemination of this "miracle cure" accelerated following the publication of peer-reviewed articles, proving the effectiveness of chlorine dioxide on COVID-19 and the promotion on Odysee of very high success rates for patients treated by physicians. This shows that physicians, as well as famous or influential people, may generate a "super influencer effect" in medical misinformation that potentiates to the infodemic problem.

This study also shows that the process of misinformation entered the sphere of scientific controversy. When scientists have unverified beliefs and try to publish them, the beliefs become potential knowledge until the process of replication allows the scientific community to decide. This allows unverified knowledge to spread widely on social networks. The boundary between science and misinformation is becoming blurred to the point that it is no longer possible until proven otherwise to call chlorine dioxide a "false miracle cure" but a controversial treatment against COVID-19.

Figure 1 .

 1 Figure 1. Comparison of the normalized number of French newspaper articles, Google requests by French users, and French messages posted on Twitter between 2006 and 2021. Sources: Europresse, Twitter API V2, and Google Trends.

Figure 1 .

 1 Figure 1. Comparison of the normalized number of French newspaper articles, Google requests by French users, and French messages posted on Twitter between 2006 and 2021. Sources: Europresse, Twitter API V2, and Google Trends.

Figure 2 .

 2 Figure 2. A total of 100 most influential tweets mentioning "chlorine dioxide" and "COVID" in French messages posted on Twitter as a function of the date and "diffusion metric."

Figure 2 .

 2 Figure 2. A total of 100 most influential tweets mentioning "chlorine dioxide" and "COVID" in French messages posted on Twitter as a function of the date and "diffusion metric".

Figure 3 .

 3 Figure 3. Sentiments (pro, con, neutral) of the 100 most influential tweets mentioning the terms "chlorine dioxide" and COVID on Twitter.

Figure 3 .

 3 Figure 3. Sentiments (pro, con, neutral) of the 100 most influential tweets mentioning the terms "chlorine dioxide" and COVID on Twitter.

Figure 4 .

 4 Figure 4. Network graph of tweets and retweets mentioning the terms "miracle cure" (N = 1252) as a function of content type (color group) and eigenvector centrality (size of nodes), using Gephi 0.9.2 and the Force Atlas algorithm.

Figure 4 .

 4 Figure 4. Network graph of tweets and retweets mentioning the terms "miracle cure" (N = 1252) as a function of content type (color group) and eigenvector centrality (size of nodes), using Gephi 0.9.2 and the Force Atlas algorithm.

Table 1 .

 1 Names mentioned in the most influential tweets.

	Name	Number of Mentions	%
	Manuel Aparicio-Alonso	9	36%
	Andreas Kalcker	7	28%
	Chinda Brandolino	3	12%
	Astrid Stuckelberger	2	8%
	Carlos Alvarado	1	4%
	Denis Agret	1	4%
	Mark Grenon	1	4%
	Patricia Callisperis	1	4%

Table 1 .

 1 Names mentioned in the most influential tweets.

	Name	Number of Mentions	%
	Manuel Aparicio-Alonso	9	36%
	Andreas Kalcker	7	28%
	Chinda Brandolino	3	12%
	Astrid Stuckelberger	2	8%
	Carlos Alvarado	1	4%
	Denis Agret	1	4%
	Mark Grenon	1	4%
	Patricia Callisperis	1	4%

Table 2 .

 2 Content

	Tweet Content	Number of Tweets	%
	Effective treatment	50	50%
	Scientific evidence	15	15%

of the 100 most influential tweets mentioning the terms "chlorine dioxide" and "COVID" on Twitter.

Table 2 .

 2 Content of the 100 most influential tweets mentioning the terms "chlorine dioxide" and "COVID" on Twitter.

	Tweet Content	Number of Tweets	%
	Effective treatment	50	50%
	Scientific evidence	15	15%
	Fact-check	14	14%
	Other	7	7%
	Political or judicial information	6	6%
	Product feedback	4	4%
	Product danger	3	3%
	Genesis II Church of Health and Healing	1	1%

Funding: This research received no external funding.

Data Protection Regulation:

The study is registered with the French National Centre for Scientific Research (CNRS) data protection officer under the following number: 2-22120.

Data Availability Statement: Data are available by contacting the corresponding author upon reasonable request. Note that, in accordance with Twitter's terms of use under the European General Data Protection Regulation, tweets cannot be shared (Twitter Controller-to-Controller Data Protection Addendum n.d.).

Conflicts of Interest:

The author declares no conflict of interest.