Florin Belgun
email: belgun@math.uni-leipzig.de

Andrei Moroianu

HOLONOMY OF TAME WEYL STRUCTURES

Keywords: 2000 Mathematics Subject Classification: Primary 53A30, 53C05, 53C29 Weyl structures, reducible holonomy, cone metrics, tame connections

.

Introduction

The holonomy of Riemannian manifolds is an extensively studied topic for a long time, and it is a basic fact that if the manifold is not locally symmetric, then the local holonomy group either acts reducibly (in which case the manifold locally splits as a Riemannian product and the holonomy group is the product of the holonomy groups of the factors), or it belongs to the Berger list [START_REF] Besse | Einstein manifolds[END_REF].

On a conformal manifold (M, c), the rôle of the Levi-Civita connection is played by the family of compatible Weyl structures, which are conformal, torsion-free connections on the tangent bundle T M [15]. Weyl structures can be closed or exact, i.e. locally, resp. globally equal to the Levi-Civita connection of some Riemannian metric in the conformal class, or non-closed.

As a consequence of the Merkulov-Schwachhöfer classification of groups occurring as holonomy of torsion-free connections [START_REF] Merkulov | Classification of irreducible holonomies of torsion-free affine connections[END_REF], the holonomy group of every non-closed irreducible Weyl structure is the full conformal group in dimensions other than 4. In [START_REF] Belgun | Weyl-parallel forms and conformal products[END_REF] we show that the reducible case is very interesting and, so far, little understood: The holonomy reduction defines locally a conformal product structure, and the holonomy group, although included in a product group, is not necessarily a product itself. In short, the reduced holonomy of a non-closed Weyl structure is either trivial, the full conformal group, some special groups in dimension 4, or it is reducible (in which case no complete description exists yet).

In contrast to that, the restricted holonomy of a closed Weyl structure is always a Riemannian holonomy (see Remark 2.4 below). However, not every Riemannian holonomy group occurs as holonomy of a closed, non-exact Weyl structure. More precisely, we show in Section 5.2 that the locally symmetric case and the quaternion Kähler holonomy Sp k • Sp 1 do not occur, while all other holonomy groups in the Berger list occur as restricted holonomy groups of closed, non-exact Weyl structures. Moreover, one can even realize them on compact manifolds, by means of the cone construction, cf. Proposition 5.8 for details.

On the other hand, no compact examples of closed, non-exact Weyl structures with reducible holonomy are known so far, besides the flat ones. In the main result of this paper (Theorem 4.1), we show that, under a certain restriction, called tame (see below), the holonomy of a closed, non-exact, Weyl structure on a compact conformal manifold is either discrete or irreducible.

It is remarkable that an analogous statement has no chance to hold for exact or nonclosed Weyl structures. Simple counter-examples are Riemannian products for the first case and conformal products (see [START_REF] Belgun | Weyl-parallel forms and conformal products[END_REF]) for the second one. In order to understand why this result is expectable for closed, non-exact Weyl structures, we consider it from the point of view of a classical result by S. Gallot [START_REF] Gallot | Équations différentielles caractéristiques de la sphère[END_REF], which states that the metric cone over a complete Riemannian manifold is either irreducible or flat.

Recall that a metric cone over a Riemannian manifold (N, g N ) is N := R * + × N, with the metric g N := dt 2 + t 2 g N , and it carries a global flow of homotheties (the flow of the radial vector field t(∂/∂t)). The metric g N and its Levi-Civita connection define by projection a conformal structure and a closed, non-exact canonical Weyl structure on the quotient of the cone by the discrete group Γ generated by one of these homotheties. Gallot's result can thus be restated as follows: If N is complete, then the canonical Weyl structure on N /Γ is irreducible or flat.

We generalize this construction in Section 2. To every closed, non-exact Weyl structure D on a conformal manifold (M, c), we associate its minimal Riemannian cover (M 0 , g 0 ), with the property that the deck transformation group acts on M 0 by strict homotheties, and the pull-back of D to M 0 is the Levi-Civita connection of g 0 . We obtain in this way a one-to-one correspondence between closed, non-exact, Weyl structures on compact conformal manifolds and incomplete Riemannian manifolds carrying a co-compact group Γ of strict homotheties acting freely and properly discontinuously, called cone-like manifolds (Remark 2.4).

In the first result of this paper, Theorem 2.6, we show that every cone-like manifold (M 0 , g 0 ) can be completed as a metric space by adding exactly on point ω, called the singularity of (M 0 , g 0 ), which emphasizes again the similarity with a cone. The crucial point of the proof is Lemma 2.8, which states that on a cone-like manifold, the distance from a fixed point to its image through any contracting homothety in Γ is bounded (a fact which does not necessarily hold on the universal covering of M 0 ).

If the restricted Riemannian holonomy of a cone-like manifold (M 0 , g 0 ) is reducible, our main result (Theorem 4.1) states that, under the tame assumption, the metric g 0 is flat. The idea of the proof is to show that the sets of leaves of any of the two integrable foliations (corresponding to the parallel splitting of the tangent bundle) contain "large" families of complete (immersed) submanifolds, all isometric to each other. On the other hand, we also show that the homotheties of (M 0 , g 0 ) preserve these families, and we end up with pairs of complete submanifolds which are at the same time isometric and homothetic to each other, thus flat.

Roughly, these ideas are inspired by the original proof of Gallot [START_REF] Gallot | Équations différentielles caractéristiques de la sphère[END_REF] of the irreducibility of the cone over a complete manifold. However, in our more general cone-like setting, the difficulty comes from the lack of information about the incomplete geodesics (which, for a cone, are just its rays, the orbits of the homothety flow). The notion of a tame connection, which we introduce in Section 3, is equivalent to the existence of uniform bounds for the life-times of the incomplete geodesics generated by vectors belonging to any compact subset of the tangent bundle, and allows us to construct the large families discussed above. Theorem 4.1 applies to a wide class of Weyl structures: We show in Section 5 that the tame condition is fulfilled by any small deformations of a cone metric, more precisely by a C 1 -neighbourhood of the canonical Weyl structure on the quotient of a cone by one of its homotheties. This is a consequence of a more general result (Theorem 5.4), which states that every Weyl structure is tame, provided that it satisfies an analytic (open) condition, called analytically tame, which is tautologically fulfilled by the canonical Weyl structure on cone quotients.

We classify, at last, all possible restricted holonomy groups of closed tame Weyl structures (Propositions 5.8 and 5.9).

The minimal Riemannian cover of a closed Weyl structure

In this section, (M, c) denotes a connected conformal manifold and D denotes a closed, non-exact, Weyl structure on (M, c) (see e.g. [START_REF] Belgun | Weyl-parallel forms and conformal products[END_REF] for the basic definitions). Let π : M → M be the universal cover of M, endowed with the induced conformal structure c := π * c, and Weyl derivative D := π * D. Since M is simply connected, D is exact, so M carries a Riemannian metric g0 ∈ c, unique up to a multiplicative constant, whose Levi-Civita covariant derivative is just D. Proof. Every element α ∈ A is a conformal transformation of ( M, c), so there exists a positive function ρ such that α * g0 = ρ 2 g0 . On the other hand, α preserves D, so the Riemannian metric α * g0 is D-parallel, therefore ρ is constant.

For every α ∈ A we denote by ρ(α) the constant of homothety. Consider the subgroup of isometric deck transformations of ( M , g0 ):

I := {α ∈ A | ρ(α) = 1}.
Of course, ρ being a group homomorphism from (A, •) to (R * + , ×), I is a normal subgroup of A. The quotient manifold M 0 := M /I is a Galois covering of M with Abelian deck transformation group Γ := A/I, isomorphic to the subgroup ρ(A) of (R * + , ×). Moreover g0 projects to a Riemannian metric g 0 on M 0 . Clearly ρ descends to a group homomorphism, also denoted by ρ : Γ → R * + , such that f * g 0 = ρ(f ) 2 g 0 for every f ∈ Γ. The pull-back of D to M 0 (still denoted by D) is the Levi-Civita connection of g 0 , and the deck transformation group Γ acts by pure homotheties on (M 0 , g 0 ) (i.e. the only isometry in Γ is the identity). This motivates the following: Definition 2.2. Let D be a closed Weyl structure on a connected conformal manifold (M, c). The triple (M 0 , g 0 , Γ), together with the covering π :

M 0 → M = M 0 /Γ is called the minimal Riemannian cover of (M, c, D).
Notice that there is no canonical way to choose g 0 in its homothety class, but all the properties we will consider in the sequel will not depend on such a choice.

If d denotes the geodesic distance on M 0 induced by the Riemannian metric g 0 , every f ∈ Γ is a homothety of the metric space (M 0 , d), i.e. d(f (x), f (y)) = ρ(f )d(x, y) for each x, y ∈ M 0 . Definition 2.3. A cone-like space is a locally compact metric space (M 0 , d) together with a finitely generated, non-trivial group Γ acting freely and properly discontinuously by homotheties on (M 0 , d), such that Γ contains no isometry besides the identity, and such that the quotient M 0 /Γ is a compact topological space.

Remark 2.4. The above considerations show that the minimal Riemannian cover defines a one-to-one correspondence between the set of triples (M, c, D) consisting in a compact manifold M, a conformal structure c and a closed, non-exact Weyl structure D on it, and the set of cone-like Riemannian manifolds (M 0 , g 0 , Γ) (modulo constant rescalings of the metric g 0 ).

A fundamental example, which is the Leitfaden of our present study, is the following: Example 2.5. Let (N, g N ) be a complete Riemannian manifold and let (M 0 , g 0 ) := (R * + × N, dt 2 + t 2 g N ) be the metric cone over N (note that g 0 and the product metric g on M 0 ≃ R × N are conformally related by setting t = e s , t ∈ R * + , s ∈ R). The multiplication by some λ > 1 on the R-factor is a strict homothety of g 0 and an isometry of g. It generates a group Γ acting freely and properly discontinuously on M 0 . The metric g projects to the product metric, also denoted by g, on the quotient manifold

M := M 0 /Γ ≃ S 1 × N. The Levi-Civita connection D 0 of g 0 is Γ-invariant, inducing therefore a closed, non- exact Weyl structure D on (M, [g]).
It is straightforward to check that (M 0 , g 0 ) is the minimal Riemannian cover of (M, [g], D). A slightly more general example of Weyl manifold having (M 0 , g 0 ) as minimal Riemannian cover can be constructed by projecting the Levi-Civita connection of g 0 onto the mapping torus of an isometry of N and, more generally, on any compact quotient of M 0 ≃ R × N by an Abelian group acting by isometries (and preserving the corresponding splitting) of the product metric.

Metric cones can be equivalently characterized by the existence of a global homothetic gradient flow, i.e. a complete vector field which is locally a gradient (with respect to a local D-parallel metric g 0 ), and acts infinitesimally by homotheties of g 0 . We will exhibit in this section some further properties which the class of cone-like spaces shares with the (much more restricted) class of metric cones.

The terminology in Definition 2.3 is justified by the following result which shows that (M 0 , d) can be completed by adding one single point.

Theorem 2.6. Let (M 0 , d, Γ) be a cone-like space. Then the metric completion of

(M 0 , d) is a metric space M 0 such that M 0 M 0 is a single point ω, called the singularity of M 0 .
Proof. Since every commutator of Γ is an isometry of (M 0 , d), the hypothesis ensures that Γ is Abelian. We need to show that (M 0 , d) contains at least one non-convergent Cauchy sequence, and that any two such sequences are equivalent.

Let f ∈ Γ be any element with ρ(f ) < 1. For every x ∈ M 0 and m < n ∈ N we have

d(f m (x), f n (x)) ≤ n-1 k=m d(f k (x), f k+1 (x)) = d(x, f (x)) n-1 k=m ρ(f ) k < d(x, f (x)) ρ(f ) m 1 -ρ(f ) ,
thus showing that {f n (x)} is a Cauchy sequence. If this sequence had a limit l in M 0 , then l would be a fixed point of f , contradicting the fact that Γ acts freely. Thus (M 0 , d) is non-complete.

Lemma 2.7. Let {x n } be a non-convergent Cauchy sequence in (M 0 , d). Then there exists x ∈ M 0 and a sequence {f n } of elements of Γ satisfying lim n→∞ ρ(f n ) = 0 such that {x n } is equivalent to {f n (x)}.

Proof. Let π denote the projection of M 0 onto the compact space M := M 0 /Γ. By choosing a subsequence if necessary, we may assume that π(x n ) converges to some y ∈ M. Take x ∈ π -1 (y). Since Γ acts properly discontinuously, there exists some open neighbourhood U 0 of x such that h(U 0 ) ∩ U 0 = ∅ for every h ∈ Γ different from the identity. We choose r > 0 such that the ball B x (2r) of radius 2r in x lies in U 0 . Then U := π(B x (r)) is a neighbourhood of y in the quotient topology, so there exists some n 0 such that π(x n ) ∈ U for n ≥ n 0 . This shows that for n ≥ n 0 there exist z n ∈ B x (r) and

f n ∈ Γ such that x n = f n (z n ).
Suppose that ρ(f n ) does not tend to zero. By taking a subsequence if necessary, we may assume that ρ(f n ) > δ for every n. For every m, n such that f n = f m , the open balls f n (B x (2r)) and f m (B x (2r)) are disjoint, being included in f n (U 0 ), and f m (U 0 ) respectively. As

f n (z n ) ∈ f n (B(x, r)) and f m (z m ) ∈ f m (B(x, r)), we get d(x n , x m ) = d(f n (z n ), f m (z m )) ≥ 2rδ.
The fact that {x n } is a Cauchy sequence ensures therefore the existence of an index N such that f n = f N for every n > N. Since B x (r) is relatively compact, we may assume (passing to some subsequence, if necessary) that z n tends to z ∈ B x (r). Thus {x n } converges to f N (z), contradicting the fact that {x n } does not converge.

This shows that lim

n→∞ ρ(f n ) = 0. Since d(f n (z n ), f n (x)) = ρ(f n )d(z n , x) < rρ(f n ),
the sequences {x n } and {f n (x)} are equivalent, thus proving the lemma.

In order to conclude the proof of the theorem we need one more technical result. Lemma 2.8. For every fixed point x ∈ M 0 there exists a constant K x , depending on x, such that d(x, f (x)) < K x for every contracting f ∈ Γ (i.e. with ρ(f ) < 1).

Proof. Let {h 1 , . . . , h n } be a system of generators of Γ with ρ i := ρ(h i ) > 1. Let D x := max {i=1,...,n} d(x, h i (x)). For every (a 1 , . . . , a k ) ∈ N k , we claim that

d x, k i=1 h a i i (x) ≤ D x k i=1 ρ a i +1 i -1 ρ i -1 . (1)
We prove the claim by induction on k. For k = 1 we have

d(x, h a 1 1 (x)) ≤ a 1 -1 s=0 d(h s 1 (x), h s+1 1 (x)) = d(x, h 1 (x)) a 1 -1 s=0 ρ s 1 ≤ D x ρ a 1 1 -1 ρ 1 -1 < D x ρ a 1 +1 1 -1 ρ 1 -1 .
Assume now that (1) holds for each k ≤ l and for every (a 1 , . . . , a k ) ∈ N k and consider some element (a 1 , . . . , a l+1 ) ∈ N l+1 . We denote by

h := l i=1 h a i i
and by y j := h j l+1 • h (x), ∀ j = 0, . . . , a l+1 .

Using (1) for k = l we have

d(x, y 0 ) ≤ D x k i=1 ρ a i +1 i -1 ρ i -1 ,
and d(y j , y j+1 ) = ρ j l+1 d(y 0 , y 1 ) = ρ(h)ρ j l+1 d(x, h l+1 (x)), which further imply

d x, l+1 i=1 h a i i (x) = d(x, y a l+1 ) ≤ d(x, y 0 ) + a l+1 -1 j=0 d(y j , y j+1 ) ≤ D x l i=1 ρ a i +1 i -1 ρ i -1 + D x l i=1 ρ a i i a l+1 -1 j=0 ρ j l+1 ≤ D x l i=1 ρ a i +1 i -1 ρ i -1 1 + a l+1 -1 j=0 ρ j l+1 ≤ D x l i=1 ρ a i +1 i -1 ρ i -1 a l+1 j=0 ρ j l+1 = D x l+1 i=1 ρ a i +1 i -1 ρ i -1 ,
thus proving our claim for k = l + 1. In order to finish the proof of the lemma, let f ∈ Γ be an element with ρ(f ) < 1. By reordering the system of generators if necessary, we can write

f = n i=1
h a i i , with a i ≥ 0 for i ≤ m and a i ≤ 0 for i ≥ m + 1.

We denote b i := -a i ≥ 0 for i ≥ m + 1. Using (1) we obtain

d(x, f (x)) = n i=m+1 ρ a i i d m i=1 h a i i (x), n i=m+1 h b i i (x) ≤ n i=m+1 ρ a i i d x, m i=1 h a i i (x) + d x, n i=m+1 h b i i (x) ≤ n i=m+1 ρ a i i D x m i=1 ρ a i +1 i -1 ρ i -1 + D x n i=m+1 ρ b i +1 i -1 ρ i -1 .
We neglect the -1 terms in the numerators above and multiply the brackets. Remembering that n i=1 ρ a i i = ρ(f ) < 1, we finally get

d(x, f (x)) ≤ D x m i=1 ρ i ρ i -1 n i=1 ρ a i i + n i=m+1 ρ i ρ i -1 ≤ D x m i=1 ρ i ρ i -1 + n i=m+1 ρ i ρ i -1 ≤ D x n i=1 ρ i ρ i -1 + 1 =: K x ,
where the last inequality follows from the fact that a + b ≤ ab + 1 for all a, b ≥ 1.

Let now {x n } be a non-convergent Cauchy sequence in M 0 . Choose y ∈ M 0 and f ∈ Γ such that ρ := ρ(f ) < 1. We claim that {x n } is equivalent to {f n (y)}. By Lemma 2.7, there exists x ∈ M 0 and a sequence {f n } of elements of Γ satisfying lim n→∞ ρ(f n ) = 0, such that {x n } is equivalent to {f n (x)}. Since lim n→∞ ρ(f n ) = 0, there exists an increasing sequence of integers {k n } such that ρ(f kn ) < ρ n . As

ρ(f -n • f kn ) < 1, Lemma 2.8 yields d(f n (x), f kn (x)) = ρ n d(x, (f -n • f kn )(x)) ≤ K x ρ n .
The sequences {f kn (x)} and {f n (x)} are thus equivalent, so the same holds for {x n } and {f n (x)}. Finally, for any y = x, {f n (x)} is clearly equivalent to {f n (y)}, thus finishing the proof of the theorem. Theorem 2.6 shows that cone-like spaces still have the one-point completion, although, in contrast to metric cones, they only carry a discrete group of homotheties. Note that the universal covering of a metric cone is a metric cone itself, therefore admits the one-point completion as well. It is unknown whether this fact holds for the universal covering of an arbitrary cone-like space (see Section 5.4).

Functions on a metric cone measuring geometric quantities like lengths, are equivariant with respect to the radial flow (acting by homotheties), and thus vary linearly on the rays.

In the more general case of cone-like spaces, we introduce, for further use, the following simple notion: Definition 2.9. Two positive functions f 1 , f 2 : M 0 → R * + are said to be equivalent if their ratio is bounded above and below by positive constants. A function which is equivalent to the distance to the singularity ω is called quasi-linear.

Denote by δ :

M 0 → R * + the distance to the singularity ω ∈ M 0 : δ(x) := d(x, ω).
Lemma 2.10. Let ψ : M 0 → R * + be any Γ-equivariant function of weight 1 on M 0 (i.e. satisfying ψ • f = ρ(f )ψ for every element f ∈ Γ), such that ψ and ψ -1 are locally bounded (e.g., ψ is continuous). Then ψ is quasi-linear.

Proof. Consider a compact fundamental domain Ω of the action of Γ on M 0 and define

k 1 := inf x∈Ω ψ(x) δ(x) , k 2 := sup x∈Ω ψ(x) δ(x) .
Because δ is continuous and ψ and its inverse are locally bounded, their quotients δ/ψ and ψ/δ are bounded on the compact set Ω. It follows that k 1 , k 2 are positive real numbers, so that

ψ(x) δ(x) ∈ [k 1 , k 2 ]
holds tautologically on Ω. Let now y be an arbitrary point of M 0 and f ∈ Γ such that x := f -1 (y) ∈ Ω. From the equivariance property of ψ we get

ψ(y) δ(y) = ψ(f (x)) δ(f (x)) = ρ(f )ψ(x) ρ(f )δ(x) = ψ(x) δ(x) ∈ [k 1 , k 2 ],
which finishes the proof.

As a consequence of the previous lemma, we show for later use that if (M 0 , g 0 ) is the minimal Riemannian cover of a closed non-exact Weyl structure D on a compact conformal manifold (M, c), then any conformal factor relating g 0 to the pull-back on M 0 of a metric in the conformal class c on M is equivalent to the distance function δ to the singularity ω ∈ M 0 : Lemma 2.11. Let g be the pull-back to M 0 of a metric in c on M and let ϕ : M 0 → R * + be defined by g 0 = ϕ 2 g. The function ϕ is then quasi-linear on M 0 .

Proof. Every element f ∈ Γ being an isometry of g, we obtain

ρ(f ) 2 ϕ 2 g = ρ(f ) 2 g 0 = f * g 0 = (ϕ • f ) 2 g, showing that ϕ • f = ρ(f )ϕ.
The assertion thus follows from Lemma 2.10.

Tame Weyl structures and their geodesics

In contrast to the Riemannian situation, a Weyl structure on a compact conformal manifold is not necessarily geodesically complete.

Example 3.1. Let (M, c) be a compact conformal manifold and let D be a closed, nonexact, Weyl structure on M. Theorem 2.6 shows that the minimal Riemannian cover (M 0 , g) of (M, c, D) is incomplete, so through every point of M 0 passes an incomplete geodesic. Its projection onto M is thus an incomplete geodesic of D.

In order to study the geometry of M 0 in the neighbourhood of its singularity ω, we need to understand the behaviour of the geodesics passing through or near ω. In principle, the dynamics of the geodesic flow of (M, g) can be rather wild near ω. Here is a list of phenomena which may occur:

(1) The lengths of the geodesics starting at some given point P and passing through ω (i.e. the life-time of an incomplete geodesic) might not be bounded. (2) There might exist closed geodesics through ω (i.e. geodesics having finite lifetime in both directions). (3) There might even exist a complete geodesic whose adherence contains ω.

3.1.

Tame connections. To begin with, let us recall some basic facts about the geodesic flow of an affine connection D on a manifold M, or, equivalently, the exponential map exp D : U → M defined on an open subset U of T M, and of regularity depending on the ones of M and D. For our purposes, we assume it is C ∞ . For X ∈ U x := U ∩ T x M, exp D (X) is the point γ(1) on the geodesic defined by γ(0) = x and γ(0) = X.

We define the life-time L D : T M → (0, +∞] of a half-geodesic generated by X ∈ T M, by L D (X) := sup{t > 0 | tX ∈ U}, in other words, the supremum of the time for which the half-geodesic tangent to X is defined. Of course, if (M, D) is geodesically complete, all life-times are infinite.

We split the complement T M {0} of the zero section in the tangent bundle into two sets, the set I D of vectors generating incomplete half-geodesics, and its complement C D . These subsets are both star-shaped, i.e. for a vector X ∈ T M

X ∈ I D ⇐⇒ sX ∈ I D , ∀ s > 0.
We exclude the zero section in T M because it generates complete geodesics, but admits a neighbourhood in C D if and only if D is geodesically complete (and thus C = T M in this case). There is nothing to say, in general, about the topology of the two subsets of the partition, as one of them is obtained as an infinite intersection of open sets. In other words, the set of vectors generating incomplete geodesics

I D is closed in T M {0}.
Example 3.3. Let (M, g) be the metric cone over a complete Riemannian manifold (N, h). The only incomplete geodesics on M are then the rays {x} × (0, t] (connecting the point (x, t) with the cone apex), and their length is t (see [START_REF] Gallot | Équations différentielles caractéristiques de la sphère[END_REF]). The Levi-Civita connection of (M, g) is thus weakly tame.

It is equally easy to construct examples of affine connections which are not weakly tame: Consider the flat torus T 2 , choose a point ω on it, and pick a dense geodesic avoiding ω. This can be seen as a limit of geodesics passing through ω. The induced flat connection on T 2 {ω} is thus not weakly tame.

Definition 3.4. A weakly tame connection

D on a manifold M is called tame if the life-time L D : I D → R * + is locally bounded.
Note that L D is lower semi-continuous on I D : If a geodesic generated by X ∈ T M is defined up to time T , all geodesics generated by vectors in some neighbourhood of X in T M are also defined up to time T .

We will see later on that the metric cone over a complete Riemannian manifold (Example 3.3) is tame.

Example 3.5. Not all weakly tame connections are tame, as the following example of a domain in R 2 shows. Take

S := {(x, y) ∈ R 2 | x > 1, x 2 y 2 < 1} {(n, 0) | n ∈ N, n > 3}
and let D be the restriction to S of the canonical flat connection in R 2 . P Figure 1. An example of weakly tame connection which is not tame.

The connection D on S is obviously weakly tame, since there are no complete halfgeodesics. On the other hand, the life-time of a half-geodesic from any point P ∈ S on the x-axis gets arbitrarily large as its defining vector approaches (1, 0) ∈ T P S (for points Q ∈ S away from the x-axis, the length of any geodesic through Q is bounded by a constant depending on Q). Therefore (S, D) is not tame. Proof. This condition easily follows if D is tame, because such intersections are themselves compact (as D is weakly tame, I D is closed).

Conversely, if X ∈ C D , then for any t > 0, there exists a compact neighbourhood U(t) of X in T M {0}, for which all the generated geodesics are defined at least up to time t: We choose, for example, a compact neighbourhood

V t ⊂ U of tX in T M (recall that the domain of definition U of exp is open in T M), then define U(t) := (1/t)V t .
As U(t) is compact, we get by hypothesis an upper bound K(t) for the life-time of all incomplete geodesics generated by vectors in U(t) ∩ I D . Let us fix t 0 > 0 and choose some T larger than K(t 0 ). The set U 0 := U(t 0 )∩U(T ) is again a compact neighbourhood of X, but now, the upper bound K(t 0 ) for L D on U 0 ∩ I D ⊂ U(t) ∩ I D and the lower bound T are contradictory, therefore U 0 ∩ I D is empty, which shows that C D is open and D is weakly tame.

Moreover, the hypothesis implies that L D is locally bounded, therefore D is tame as well.

Remark 3.7. In the definition of a tame structure, we ask for L D to be locally bounded and for D to be weakly tame. The following simple example, along with Example 3.5, shows that these two conditions are independent: Example 3.8. Let Z := S 1 × (a, b) be a bounded, flat cylinder. The only complete geodesics on Z are the circles, hence C D ⊂ T Z {0} is the sub-bundle of tangent lines to the geodesic circles, in particular it is not open, therefore Z is not weakly tame.

On the other hand, the length of an incomplete geodesic depends continuously on its slope with respect to the geodesic circles, therefore L D : I D → (0, ∞) is a continuous function defined on an open subset of T Z, in particular it is locally bounded.

3.2. Tame Riemannian metrics. We will now investigate the notions defined above in the particular case where D is the Levi-Civita connection of a Riemannian manifold, in particular on the minimal Riemannian cover (M 0 , g 0 ) of a closed, non-exact Weyl manifold. Note that a connection on M is (weakly) tame if and only if its pull-back to any covering of M is (weakly) tame. In order to characterize tame closed Weyl structures on a compact manifold, it thus suffices to understand tame cone-like Riemannian manifolds. In the following lemmas (that hold in a general setting), we intentionally use notations from the previous sections, to emphasize where our main interest lies: Lemma 3.9. A Riemannian manifold (M 0 , g 0 ) is weakly tame if and only if the set

I g 0 := I D ∩ S(T M 0 ) is closed.
Here D is the Levi-Civita connection of g 0 and S(T M 0 ) is the sphere bundle of unit vectors in T M 0 .

The proof is obvious. Note the slight difference between I D and I g 0 . We also denote the restriction of L D to I g 0 by L g 0 and I g 0 ∩ T x M 0 =: I g 0

x .

For an incomplete Riemannian manifold (M 0 , g 0 ), we set

µ : M 0 → (0, +∞], µ(x) := sup X∈I g 0 x L D (X).
It is the supremum of the lengths of all incomplete half-geodesics starting in x.

Lemma 3.10. The Riemannian manifold (M 0 , g 0 ) is tame if and only if µ is locally bounded.

Proof. Note first that, since M 0 is incomplete, from every point starts an incomplete half-geodesic, and so the set I g 0 x is non-empty for any x ∈ M 0 , hence µ is well-defined. If D is tame, the fact that µ is locally bounded follows immediately from Proposition 3.6.

Conversely, let K ⊂ T M 0 {0} be a compact set. Then its projection K 0 on M 0 is compact as well, and there exists q > 1 such that

K ⊂ {X ∈ T x M 0 | x ∈ K 0 , 1 q ≤ X ≤ q}.
The local boundedness of µ, together with the inequality above implies that

L D (X) ≤ q sup K 0 µ, ∀ X ∈ K ∩ I D ,
so D is tame by Proposition 3.6.

We give now a criterion characterizing closed tame Weyl structures on compact manifolds, or, equivalently, on cone-like Riemannian spaces: Proposition 3.11. Let (M, c) be a compact conformal manifold, and let D be a closed, non-exact Weyl structure on it. Then D is tame on M (or, equivalently, on M 0 ) if and only if µ : M 0 → (0, +∞] is (finite and) quasi-linear on M 0 .

Proof. The distance δ to the singularity ω ∈ M 0 is always continuous on M 0 , and µ ≥ δ, therefore µ -1 is locally bounded. On the other hand, δ and µ are clearly Γ-equivariant of weight 1, because both denote geometrical lengths. If D is tame, Lemma 2.10 implies, together with Lemma 3.10, the quasi-linearity of µ.

Conversely, if µ is quasi-linear, then it is locally bounded, therefore, again by Lemma 3.10, D is tame.

In the next section we prove our main result, concerning the holonomy of a closed tame Weyl structure. Finally, in Section 5 we will show that the tame condition applies to an open set of Weyl structures (in the C 1 -topology), in particular the class of tame closed Weyl structures is significantly large. We also give the complete classification of their possible restricted holonomy groups.

Closed tame Weyl structures with reducible holonomy

The goal of this section is to prove the following Proof. We start by showing that if the restricted holonomy Hol 0 (D) is reducible, then there exists a finite covering M of M on which the full holonomy of the pull-back of D has reducible holonomy. In order to keep the argument as simple as possible, we will not be very precise on the holonomy groups and consider them as abstract groups rather than as transformation groups of each tangent space.

Consider the metric g0 on the universal cover M of M (defined up to a multiplicative constant), whose Levi-Civita covariant derivative ∇ is the pull-back of D to M . The holonomy of ∇ is clearly equal to the restricted holonomy of D. By Theorem IV.5.4 in [START_REF] Kobayashi | Foundations of Differential Geometry I[END_REF], the tangent bundle of M splits in a direct sum T M = T 0 ⊕ . . . ⊕ T m of ∇-parallel sub-bundles and the holonomy group of ∇ satisfies Hol( ∇) = H 1 × . . . × H m , where H i acts irreducibly on T i and trivially on T j for j = i (T 0 being the flat component). Moreover this decomposition is unique up to a permutation of the set {1, . . . , m} (such permutations may occur if some of the factors H i coincide). By Lemma 2.1, every element f ∈ A of the deck transformation group of the covering M → M is affine with respect to ∇, so there exists a permutation σ f of {1, . . . , m} such that f * (T i ) = T σ f (i) . Let B ⊂ A be the kernel of the group homomorphism A → S m given by f → σ f . The metric g0 and the connection ∇ on M descend to a conformal structure c and a Weyl structure D on the quotient M := M/B, which is a finite covering of M with group A/B ⊂ S m . By construction, the holonomy group of D on M is reducible.

Replacing (M, c, D) by ( M , c, D), we can from now on assume that the full holonomy of D is reducible. This implies that the tangent bundle of the minimal Riemannian cover (M 0 , g 0 ) of (M, c, D) splits in a direct sum of orthogonal distributions

T M = V 1 ⊕ V 2 ,
parallel with respect to the Levi-Civita connection ∇ 0 = D of g 0 . These distributions are integrable, hence define two orthogonal (and complementary) foliations on M 0 .

We will use the notion maximal leaf M i , i = 1, 2, through x ∈ M 0 to denote the set of points that can be connected to x by means of a smooth curve tangent to V i . It is a standard fact that M i are immersed submanifolds of M 0 .

We start with two preliminary results which hold on every (not necessarily complete) reducible Riemannian manifold (M 0 , g 0 ).

Lemma 4.2. Let U 1 be a leaf of V 1 (not necessarily complete) and let X ∈ V 2 be a parallel vector field along U 1 . Assume that exp x tX is defined for all x ∈ U 1 and t ∈ [0, 1]. Then x → ψ(x) := exp x X maps U 1 isometrically onto its image.

Proof. The statement is classical but as we could not find a precise reference, we provide the proof for the reader's convenience.

We first show that the map ψ is a local isometry between U 1 and some other leaf

U ′ 1 of V 1 . Consider the map ϕ : U 1 × [0, 1] → M 0 defined by ϕ(x, t) := exp x tX. Define X (x,t) ∈ T ϕ(x,t) M 0 by X (x,t) := d ds s=t ϕ(x, s).
In other words, X (x,t) is the tangent vector to the geodesic s → exp x sX at s = t, so we clearly have the relation ϕ(x, t + s) = exp ϕ(x,t) sX (x,t) .

(2) Let us fix x ∈ U 1 and denote x t := ϕ(x, t). The local de Rham decomposition theorem (Proposition IV.5.2 in [START_REF] Kobayashi | Foundations of Differential Geometry I[END_REF]) states that each x t has a neighbourhood U(t) in M 0 isometric to a Riemannian product U(t) ≃ U 1 (t) × U 2 (t), where U 1 (t) and U 2 (t) are local leaves of V 1 and V 2 through x t .

The geodesic segment ϕ({x} × [0, 1]) is compact, so it can be covered by a finite number of neighbourhoods U 2 (s 1 ), . . . , U 2 (s n ), with 0 = s 1 < . . . < s n = 1. Choose now

t i ∈ (s i , s i+1 ) ∀ i = 1, . . . , n -1, such that ϕ(x, t i ) ∈ U 2 (s i ) ∩ U 2 (s i+1
), and set t n := 1. For k = 1, . . . , n, let V k be the open subset of U 1 defined by

V k := {y ∈ U 1 | ϕ(y, s k ) ∈ U(s k )}.
We denote by V the intersection of the V k 's and by W k the subset of U(t k ) given by

W k := ϕ(V × {t k }). X k+1 U(s k+2 ) X k U(s k ) W k U 1 U(s k+1 ) U ′ 1 X V x 0 = x W n x 1 W k+1 Figure 2.
Stepwise exponentiation along the geodesic segment ϕ ({x} × [0, 1]).

Consider the vector field X k along W k whose value at ϕ(y, t k ) is X (y,t k ) . By construction, there exists a bijection ϕ k : W k → W k+1 defined by ϕ k (ϕ(y, t k )) := ϕ(y, t k+1 ) for all y ∈ V .

We claim that for every k = 1, . . . , n,

(1) ϕ k-1 is an isometry; (2) the vector field X k is parallel along W k ; (3) W k is an open subset of the leaf U 1 (t k ).
For k = 1, the first statement is empty, and the other two hold by hypothesis.

Assume that the claim holds for some

k ≥ 1. Since X k is parallel along W k , it is constant in the product coordinates on W k × U 2 (s k ) ⊂ U(s k ) (i.e. there exists Z ∈ T xt k U 2 (s k ) such that (X k ) (z,xt k ) = (0, Z), ∀ z ∈ W k ). By (2) we have ϕ k (ϕ(y, t k )) = exp ϕ(y,t k ) (t k+1 -t k )X (y,t k ) so in the product coordinates ϕ k (z, x t k ) = (z, x t k+1 ) for all z ∈ W k , showing that W k ⊂ U 1 (t k )
and that ϕ k is an isometry. Moreover X k+1 is constant along W k+1 in these coordinates, thus proving the induction step.

We have shown that in the neighbourhood

V of x in U 1 , the map x → ψ(x) = exp x X is the composition of n-1 isometries ϕ n-1 •• • ••ϕ 1 between local leaves of the distribution V 1 .
As this holds in the neighbourhood of every point x of U 1 , ψ is a local isometry from U 1 to its image U ′ 1 . In particular, this shows that U ′

1 is an open subset of the complete integral leaf of V 1 passing through ϕ(x, 1).

Moreover, the claim above shows that Y := ψ * (X) is a well-defined parallel vector field along U ′ 1 . Consider the map ψ : U ′ 1 → M 0 defined by y → exp y (-Y ). From the local considerations above, it is clear that ψ • ψ is the identity of U 1 . This shows that ψ is one-to-one.

The next result, which is somewhat folkloric, like the previous one, shows that exponentiating a geodesic tangent to V 1 in the direction of a constant or affine Jacobi field tangent to V 2 yields another geodesic whenever it is defined. Lemma 4.3. Let γ : [a, b] → M 0 be a geodesic tangent to V 1 parametrized by arc-length and let X ∈ T γ(a) M 0 be a vector tangent to V 2 . Extend X to a parallel vector field along γ.

(i) Assume that γ s (t) := exp γ(t) (sX) is well-defined for all t ∈ [a, b] and s ∈ [0, 1]. Then γ 1 (t) is a geodesic in M 0 and its tangent vector at t is the parallel transport of γ(t) at exp γ(t) (X) along the geodesic s → exp γ(t) (sX).

(ii) Assume that γ X (t) := exp γ(t) (tX) is well-defined for all t ∈ [a, b]. Then γ X (t) is a geodesic in M 0 and the projections of γX (t) onto V 1 and V 2 are parallel vector fields along γ X of length 1 and |X| respectively.

Proof. (i) The first statement follows immediately from Lemma 4.2. The second one is a consequence of the claim used to prove the same lemma.

(ii) Assume first that M 0 is a global Riemannian product M 0 = M 1 × M 2 . If γ(a) = (m 1 , m 2 ), then γ(t) = (γ 1 (t), m 2 )
for some geodesic γ 1 in M 1 parametrized by arc-length. The vector field X along γ can be written X = (0, X 2 ), where X 2 is a constant vector tangent to M 2 at m 2 . Denoting by γ 2 (t) = exp m 2 tX 2 the geodesic in M 2 starting at m 2 with initial speed X 2 , then γ X (t) = (γ 1 (t), γ 2 (t)), is a geodesic in M 0 . The projections of γX (t) onto V 1 and V 2 are ( γ1 , 0) and (0, γ2 ), which are clearly parallel vector fields along γ X of length 1 and |X| respectively. Back to the general case, it is of course enough to show that the statement holds in the neighbourhood of every point γ X (t 0 ). Since the domain of definition of the exponential on the normal bundle of a geodesic is open, the curve c(t) := exp γ(t) (t 0 X) is well-defined for t near t 0 . By Lemma 4.2, c(t) is a geodesic through x := γ X (t 0 ), parametrized by arc-length. Moreover, if Y denotes the parallel vector field along c(t) By the local de Rham theorem, the point x has a neighbourhood U isometric to U 1 × U 2 , where U i is some local leaf of V i through x. As γ X (t) lies in U for t near t 0 , the statement follows from the first part of the proof.

with Y x = d exp γ(t 0 ) (t 0 X), Lemma 4.2 also shows that Y γ(t) = d exp γ(t) (t 0 X), so by (2), γ X (t) = exp γ(t) ((t -t 0 )Y ). c γ(t 0 ) γ x Y U γ X (t) = exp γ(t) tX tX γ(t)
We assume from now on that D is a closed tame Weyl structure on a compact conformal manifold (M, c) which has reducible holonomy and that (M 0 , g 0 ) is the minimal Riemannian cover of (M, c, D). We denote as before by d the distance induced by g 0 on M 0 , by ω the singularity of M 0 and by δ the distance to the singularity: δ(x) := d(x, ω). Since D = ∇ 0 , g 0 has reducible holonomy, so the above results apply to the present setting. We will need the following quantitative version of the local de Rham decomposition theorem for (M 0 , g 0 ).

Lemma 4.4. There exists a quasi-linear function σ : M 0 → R * + such that each point x ∈ M 0 has a neighbourhood U and an isometry F : U → B 1

x (σ(x)) × B 2 x (σ(x)), with F (x) = (x, x), where B i x (r) is the ball of radius r around x in the maximal leaf M i though x, tangent to the distribution V i . Proof. By Lemma 2.10, it is enough to define σ on a relatively compact fundamental domain K ⊂ M 0 of the covering M 0 → M, and to extend it to M 0 in a Γ-equivariant way by σ(f (x)) = ρ(f )σ(x) for all f ∈ Γ and x ∈ K.

The local de Rham theorem ensures that for every x ∈ M 0 there exist neighbourhoods U i (x) of x in the maximal leaf M i (x) though x, tangent to the distribution V i , such that U 1 (x) × U 2 (x) is isometric to a neighbourhood U(x) of x in M 0 . Take a finite number of points x i such that K ⊂ ∪ i U(x i ). Each neighbourhood U 1 (x i ) and U 2 (x i ) contains a geodesic ball centered in x i of radius r 1 (x i ) and r 2 (x i ) respectively. It is then enough to define σ on K to be the minimum of all these radii.

We now come to a key point of the proof of Theorem 4.1, namely the existence of complete maximal leaves tangent to the distributions V i . Proposition 4.5. If M 1 is a maximal leaf of V 1 which is incomplete, then every maximal leaf of V 2 which intersects M 1 is complete.

Proof. Since M 1 is totally geodesic and incomplete, through every point x ∈ M 1 passes a geodesic γ : (0, r] → M 1 parametrized by arc-length, such that γ(r) = x, which can not be defined at t = 0. Since M i is totally geodesic in M 0 , γ is also a geodesic in M 0 . By Theorem 2.6, we must have lim t→0 γ(t) = ω in ( M 0 , d).

Let X ∈ T x M ∩ V 2 be any unit normal vector to M 1 at x, extended as before to a parallel vector field along γ. We claim that the geodesic generated by X in M 0 is complete.

The crucial point here is the fact that every point γ(t) is far enough from the singularity ω, in order to ensure that the exponential function is well-defined in a suitable neighbourhood. More precisely, Proposition 3.11 shows that there exists a constant κ > 0 such that for every t ∈ (0, r], the distance δ(γ(t)) from γ(t) to ω (in M 0 ) is bounded from below by κt. Consequently, exp γ(t) sX is well-defined for |s| ≤ κt, so by Lemma 4.3 (ii), the curve γ 1 : (0, r] → M 0 defined by γ 1 (t) := exp γ(t) κtX is a geodesic in M 0 with | γ1 | 2 = 1 + κ 2 . Moreover, the limit in M 0 of γ 1 (t) as t → 0 is clearly ω. Proposition 3.11 applied this time to the geodesic parametrized by arc-length γ1 defined by

γ1 (t) := γ 1 ((1 + κ 2 ) -1/2 t) yields δ(γ 1 (t)) > κt, whence δ(γ 1 (t)) > (1 + κ 2 ) 1/2 κt > κt.
Consequently, for every t ∈ (0, r], every geodesic defined by a unit vector Y ∈ T γ 1 (t) M 0 is defined at least up to the time κt. Taking Y to be the speed vector of the geodesic s → exp γ(t) sX at s = κt, we obtain that this geodesic can actually be extended for s ∈ [0, 2κt], for any t ∈ (0, r]. By Lemma 4.3 (ii), the curve γ 2 : (0, r] → M 0 defined by γ 2 (t) := exp γ(t) 2κtX is thus a geodesic in M 0 with | γ2 | 2 = 1 + 4κ 2 . Again, we check that the distance from γ 2 (t) to the singularity is at least κt, showing that for every t ∈ (0, r], exp γ(t) sX is well-defined for |s| ≤ 3κt. Iterating the same argument shows that the geodesic exp γ(t) sX is actually defined for every t ∈ (0, r] and for every s ∈ R. In particular, for t = r, γ(t) = x, we have proved that the geodesic through x tangent to X ∈ V 2 is complete. Since X was arbitrarily chosen, the whole integral leaf of V 2 through x is thus complete.

γ 2 (t) = exp γ(t) 2κtX γ 3 (t) = exp γ(t) 3κtX γ 1 (t) = exp γ(t) κtX γ ω
In order to apply this result, we need to show that incomplete leaves actually exist. Lemma 4.6. There exist incomplete maximal leaves M i of V i or, equivalently, incomplete geodesics γ i tangent to V i for i = 1 and i = 2.

Proof. Let γ : (0, 1] → M 0 be an incomplete geodesic, such that lim t→0 γ(t) = ω in ( M 0 , d). We may assume that γ is not tangent to V 1 or V 2 : If for instance γ were tangent to V 1 , we replace it by γ X given by Lemma 4.3 (ii), which is neither tangent to

V 1 nor to V 2 .
Let X 1 and X 2 denote the projections of γ on V 1 and V 2 respectively, which are clearly parallel along γ. We denote by r i := |X i | = 0 the norms of X i and by r := r 2 1 + r 2 2 the norm of γ. Define the slope of γ to be the quotient q(γ) := r 1 /r 2 .

We claim that γ 1 (t) := exp γ(t) (-tX 1 ) is defined for every t ∈ (0, 1], and is an incomplete geodesic tangent to V 2 , such that lim t→0 γ 1 (t) = ω in ( M 0 , d). The argument is similar to that used in the proof of Proposition 4.5: The exponential, denoted γ s (t) of -tsX 1 at γ(t) is well-defined by Proposition 3.11 for |s| ≤ κr r 1 . For every fixed s in this interval, γ s (t) is an incomplete geodesic and its slope is (see Lemma 4.3

(ii)) q(γ s ) = (1 -s)q(γ). If κr r 1 ≥ 1, which is equivalent to q(c) ≤ (κ -2 -1) -1/2
, the incomplete geodesic γ s has zero slope for s = 1, i.e. it is tangent to V 2 . Otherwise, we replace γ by γ s with s = κr r 1 and repeat this procedure. The slope of the new geodesic is

q(γ s ) = 1 - κr r 1 q(γ) = q(γ) - κr r 2 ≤ q(c) -κ,
showing that the procedure stops after a finite number of iterations. Since V 1 and V 2 play symmetric rôles, this finishes the proof.

Corollary 4.7. If there exists an incomplete geodesic γ passing through a point x ∈ M 0 such that γ is neither tangent to V 1 nor to V 2 , then x belongs to a complete leaf of V 1 which intersects an incomplete maximal leaf of V 2 , and to a complete leaf of V 2 which intersects an incomplete maximal leaf of V 1 .

Proof. The result follows directly from the proof of Lemma 4.6 together with Proposition 4.5.

Lemma 4.8. If M 1 is a maximal leaf of V 1 which is incomplete, then all maximal leaves of V 2 which intersect M 1 are isometric.
Proof. Let M 2 (x) denote the maximal leaf of V 2 through x. Since every two points of M 1 can be joined by a broken geodesic, it is enough to show that M 2 (γ(0)) and M 2 (γ(1)) are isometric for every geodesic γ : [0, 1] → M 1 . Consider the normal vector field X(t) on M 2 (γ(t)) obtained by parallel transport of γ(t). Since M 2 (γ(0)) is complete, every point y ∈ M 2 (γ(0)) can be expressed as y = exp γ(0) (Y ) for some Y ∈ T γ(0) M 2 (γ(0)). We extend Y along γ by parallel transport. By Proposition 4.5, the leaves M 2 (γ(t)) are complete, hence exp γ(t) (sY ) is well-defined for every s ∈ R and t ∈ [0, 1]. By Lemma 4.3 (i) we get exp γ(t) (Y ) = exp y (tX(0)). The exponential of X(0) is thus defined for all y ∈ M 2 (γ(0)) so we conclude by Lemma 4.2.

Consider now a metric g on M 0 obtained as the pull-back of a metric in the conformal class c on M. Let ϕ be the conformal factor relating g to g 0 by g 0 = ϕ 2 g and let d the geodesic distance induced on M 0 by g. Denote by B x (r) and Bx (r) the set of points at distance less than r from x with respect to d and d respectively. Recall that by Lemma 2.11 ϕ is quasi-linear, so there exist positive constants k 1 , k 2 such that Proof. For every y ∈ B x (r) we have δ(y) ≤ δ(x) + r, so by ( 3) ϕ(y) ≤ k 2 (δ(x) + r).

k 1 δ(x) ≤ ϕ(x) ≤ k 2 δ(x), ∀ x ∈ M 0 . (3
Consequently, the g 0 -length l 0 (c) and g-length l(c) of every path contained in B x (r) are related by l 0 (c) ≤ k 2 (δ(x) + r)l(c).

(4) Assume there exists z ∈ Bx (r) B x (r) and let c : [0, 1] → M 0 be any path joining x and z. Define s 0 = inf{s | c(s) / ∈ B x (r)} and consider the path c ′ = c| [0,s 0 ] which is clearly contained in B x (r). Then l 0 (c ′ ) ≥ r, so by (4), l(c) ≥ l(c ′ ) ≥ r. Since this holds for every path c, we must have d(x, z) ≥ r, contradicting the fact that z ∈ Bx (r). Lemma 4.10. Let γ : (0, a] → M 0 be an incomplete g 0 -geodesic parametrized by arclength, such that lim t→0 δ(γ(t)) = 0. There exist positive real numbers ρ, q ∈ (0, 1) such that the open balls B n := B γ(q n ) (ρq n ), n ∈ N, are all pairwise disjoint. Proof. Recall that by Proposition 3.11 we have control on the distance from γ(t) to the singularity ω, i.e. there exists a constant κ ∈ (0, 1), independent of γ, such that:

κt ≤ δ(γ(t)) ≤ t, ∀ t ∈ (0, a]. (5)
We start with arbitrary ρ and q in (0, 1). For every y ∈ B n , Equation ( 5) yields (κρ)q n ≤ δ(γ(q n ))ρq n ≤ δ(y) ≤ δ(γ(q n )) + ρq n ≤ (ρ + 1)q n , so by (3) we get k 1 (κρ)q n ≤ ϕ(y) ≤ k 2 (ρ + 1)q n . It is thus enough to choose ρ and q such that k 2 (ρ + 1)q n+1 < k 1 (κρ)q n for every n, which is equivalent to ρ < κ and q

< k 1 k 2 (ρ+1) (κ -ρ). Corollary 4.11. Consider the open subset B := ∪ n≥1 B n in M 0 . There exists f ∈ Γ different from the identity such that f (B) ∩ B = ∅.
Proof. Lemma 4.9 applied to x = γ(q n ) and r = ρq n shows that B n contains the open ball Bx (r) where

r = r k 2 (δ(x) + r) = ρq n k 2 (δ(x) + ρq n ) ≥ ρq n k 2 (q n + ρq n ) = ρ k 2 (1 + ρ) .
Recall that Γ acts by isometries on (M 0 , g) and that (M 0 , g)/Γ = (M, g). If f (B n )∩B m = ∅ for every f ∈ Γ and m = n, the projections π(B n ) of B n onto M would be pairwise disjoint sets, each of them containing a ball of g-radius ρ k 2 (1+ρ) in M. This is impossible since M is compact, thus proving our assertion. The last step in the proof of Theorem 4.1 is the following: Lemma 4.12. Let M 1 be a maximal leaf of V 1 which is incomplete. Then for every x ∈ M 1 , the maximal leaf M 2 (x) of V 2 through x is flat.

Proof. Let γ : (0, 1] → M 1 be an incomplete geodesic with respect to g 0 parametrized by arc-length, such that γ(1) = x and γ(t) converges to ω (with respect to d) as t tends to 0. By Lemma 4.10 one can find ρ, q ∈ (0, 1) such that the open balls B n := B γ(q n ) (ρq n ) are pairwise disjoint. Moreover, one can choose ρ such that each maximal leaf of V 2 through a point of B = ∪ n≥1 B n intersects M 1 . Indeed, this follows from Lemma 4.4 provided that ρq n is smaller than σ(γ(q n )) for every n. Since σ is quasi-linear, there exists some σ 0 such that σ(x) ≥ σ 0 δ(x), so from (5) it suffices to take ρ < κσ 0 . Corollary 4.11 now shows that there exists f ∈ Γ different from the identity and y, z ∈ B such that y = f (z). Then f maps the integral leaf M 2 (z) of V 2 through z to the integral leaf of V 2 through y. Since both leaves intersect M 1 , Lemma 4.8 shows that they are isometric. Composing f with this isometry we obtain a strict homothety of M 2 (z). Since M 2 (z) is complete, Lemma 2, page 242 in [START_REF] Kobayashi | Foundations of Differential Geometry I[END_REF] shows that it must be flat. By Lemma 4.8 again, all the other leaves tangent to V 2 must be flat as well.

We are now in position to complete the proof of Theorem 4.1. From Corollary 4.7, and Lemma 4.12, the sectional curvature of g 0 vanishes at each point x which belongs to an incomplete geodesic which is neither tangent to V 1 nor to V 2 . The proof of Lemma 4.3 (ii) shows that the set of such points is dense in M 0 . Thus (M 0 , g 0 ) is a flat Riemannian manifold, so the holonomy group of D = ∇ 0 is discrete.

Remark 4.13. The only place where the compactness assumption on M is needed in Theorem 4.1, is to ensure, by Theorem 2.6, that the minimal Riemannian cover of (M, c, D) has exactly one singularity. Theorem 4.1 thus holds in a slightly more general setting, and applies in particular to all metric cones over complete Riemannian manifolds.

Examples and applications

5.1. Analytically tame Weyl structures. In order to show that the hypotheses in Theorem 4.1 are satisfied by a large variety of Weyl structures, we introduce the following: Definition 5.1. A Weyl structure D on a conformal manifold (M, c) is called analytically tame if there exists a complete Riemannian metric g ∈ c and a positive real number ε > 0 such that

|θ| 2 g(X, X) + (∇ X θ)(X) ≥ 2εg(X, X), ∀ X ∈ T M, (6)
where ∇ denotes the Levi-Civita covariant derivative of g and θ denotes the Lee form of D with respect to g.

Recall that the Lee form θ of a Weyl structure D with respect to a metric g ∈ c measures the difference between D and the Levi-Civita connection ∇ = ∇ g of g:

D X Y -∇ X Y = θX (Y ) := θ(X)Y + θ(Y )X + θ ♯ g(X, Y ), ∀ X, Y ∈ T M, (7)
where θ = g(θ ♯ , •).

Example 5.2. With the notations form Example 2.5, it is easy to see that the standard Weyl structure D 0 on (a compact quotient of) a metric cone is analytically tame. Indeed, the Lee form of D 0 with respect to the complete metric g is θ 0 := ds and it is parallel for ∇ = ∇ g . Therefore |θ 0 | 2 g + ∇θ = g thus ( 6) is even an equality for ε = 1/2.

Remark 5.3. An exact Weyl structure on a compact conformal manifold M can not be analytically tame. Indeed, its Lee form θ with respect to any metric is exact, θ = dϕ so ( 6) cannot hold at points where ϕ reaches its maximum on M.

The remaining part of this section is devoted to the proof of the following result, which also justifies the terminology in Definition 5.1: Theorem 5.4. If D is an analytically tame Weyl structure with respect to some complete metric g ∈ c on a conformal manifold (M, c), then D is tame.

Proof. Let θ be the Lee form of D with respect to g. If ∇ denotes the Levi-Civita covariant derivative of g, then we get from [START_REF] Gauduchon | La 1-forme de torsion d'une variété hermitienne compacte[END_REF], see also [START_REF] Gauduchon | Structures de Weyl-Einstein, espaces de twisteurs et variétés de type S 1 × S 3[END_REF]:

D X -∇ X = (θ ∧ X) * -pθ(X)Id (8)
on T * M ⊗p , where (θ ∧ X) * is the usual extension of the endomorphism θ ∧ X as a derivation (in fact, the right hand side of ( 8) is just θX , acting as a derivation on T * M ⊗p ). It is easy to check that (θ ∧ X) * g = 0, so (8) yields

D X g = -2θ(X)g. (9)
On the other hand, applying [START_REF] Gauduchon | Structures de Weyl-Einstein, espaces de twisteurs et variétés de type S 1 × S 3[END_REF] to the Lee form θ itself yields

D X θ = ∇ X θ + g(θ, θ)g(X, .) -2θ(X)θ, thus showing that (6) is equivalent to (D X θ)(X) ≥ 2εg(X, X) -2θ(X) 2 , ∀ X ∈ T M. (10)
Let γ(t) be a geodesic with respect to D on M and let I = (a, b) denote its maximal domain of definition, with a, b ∈ R = R ∪ {±∞}. We introduce the functions F (t) := g( γ(t), γ(t)) -1 2 and H(t) := θ( γ(t)), defined on I. Using (9) we get

F ′ (t) = γ(t).F (t) = - 1 2 (D γ(t) g)( γ(t), γ(t)) g( γ(t), γ(t)) -3 2 = F (t)H(t), (11)
and from [START_REF] Kobayashi | Foundations of Differential Geometry I[END_REF],

H ′ (t) = γ(t).H(t) = (D γ(t) θ)( γ(t)) ≥ 2εF (t) -2 -2H(t) 2 . ( 12
)
Lemma 5.5. If b < ∞ (i.e. γ is incomplete toward the future) then lim t→b F (t) = 0.

Similarly, if a > -∞, then lim t→a F (t) = 0.

Proof. We have to show that the g-norm of the speed vector of an incomplete D-geodesic cannot be bounded on M. Consider the geodesic flow of D, viewed as a vector field on the tangent bundle T M. Let γ be the maximal half-geodesic with respect to D, issued from some X ∈ T x M. There exists T > 0 such that the maximal integral curve through X of the geodesic flow is defined only for t < T . Assume that the g-norm of γ is bounded: g( γ(t), γ(t)) < k 2 for all t ∈ [0, T ). Then the corresponding integral curve is contained in the subset K(k, T ) of T M defined by

K(k, T ) := {Y y ∈ T M | d(x, y) ≤ kT and g(Y, Y ) ≤ k 2 },
where d denotes the geodesic distance with respect to g. Since g is complete, the closed geodesic balls are compact, so K(k, T ) is a compact subset of T M. On the other hand, it is well-known that an incomplete integral curve of a vector field cannot be contained in any compact subset, thus proving the lemma.

In order to fix the ideas, we will assume from now on that 0 ∈ I and H(0) ≤ 0

(this can always be achieved by making a translation in time and replacing γ(t) with γ(-t) if necessary).

Lemma 5.6. The function F has at most one critical point. If this happens, then the critical point is an absolute minimum of F , and γ is complete. Conversely, if γ is complete, then F has a critical point.

Proof. Let t 0 be a critical point of F . From [START_REF] Lebrun | Anti-self-dual Hermitian metrics on blown-up Hopf surfaces[END_REF], H(t 0 ) = 0. Moreover, [START_REF] Merkulov | Classification of irreducible holonomies of torsion-free affine connections[END_REF] shows that H ′ is strictly positive at each point where H vanishes, so actually H cannot vanish more than once. Thus H(t) is positive for t ≥ t 0 and negative for t ≤ t 0 , so t 0 is a global minimum of F . Lemma 5.5 then shows that γ cannot be incomplete.

Conversely, assume that γ is complete, i.e. I = R. If F has no critical point, H does not vanish, so by our assumption [START_REF] Tricceri | Some examples of locally conformal Kähler manifolds[END_REF], H < 0 on R. F is thus a decreasing positive function, so lim t→∞ H(t)F (t) = lim t→∞ F ′ (t) = 0. Dividing by H(t) 2 in (12) yields

H ′ (t) H(t) 2 > 2ε F (t) 2 H(t) 2 -2.
Since the right hand side tends to infinity as t → ∞, an integration shows that lim t→∞ H(t) = 0. Using (12) again, we then see that there exist some t 0 ∈ R and δ > 0 such that H ′ (t) > δ for t > t 0 . This of course contradicts the fact that H is negative on the whole real line R, thus proving the lemma.

The convention (13) together with Lemma 5.6 ensures that if γ is an incomplete geodesic, H is negative on I, so F is decreasing. By Lemma 5.5, γ is complete toward -∞, i.e. I = (-∞, b), with b ∈ R + .

Lemma 5.7. If γ : I → M 0 is a geodesic with respect to D which is incomplete in the positive direction, then

H(t) ≤ - √ ε F (t) , ∀ t ∈ I. ( 14
)
Proof. If [START_REF] Vaisman | Locally conformal Kähler manifolds with parallel Lee form[END_REF] does not hold, there exists t 0 ∈ I such that

- √ ε F (t 0 ) < H(t 0 ) < 0.
We define the open set

I ′ := {t ∈ I | - √ ε F (t) < H(t)},
and let (a ′ , b ′ ) be the connected component of I ′ containing t 0 . By [START_REF] Merkulov | Classification of irreducible holonomies of torsion-free affine connections[END_REF], H is strictly increasing on I ′ . In the other hand, we have seen that F is decreasing on I. If b ′ ∈ I, we would have

H(b ′ ) = - √ ε F (b ′ ) < - √ ε F (t 0 )
< H(t 0 ), a contradiction. The only possibility left is thus b ′ = b. But this is impossible as well, since by [START_REF] Lebrun | Anti-self-dual Hermitian metrics on blown-up Hopf surfaces[END_REF],

lim t→b ′ log(F (t)) = log(F (t 0 )) + b ′ t 0 H(t)dt > -∞, contradicting Lemma 5.5.
Back to the proof of Theorem 5.4, using [START_REF] Vaisman | Locally conformal Kähler manifolds with parallel Lee form[END_REF] and [START_REF] Lebrun | Anti-self-dual Hermitian metrics on blown-up Hopf surfaces[END_REF] we get F ′ (t) ≤ -√ ε, and thus

√ εb = b 0 √ εdt ≤ - b 0 F ′ (t)dt ≤ F (0) = g( γ(0), γ(0)) -1 2 .
In other words, the life-time of every geodesic γ, incomplete in the positive direction, is bounded from above by (εg( γ(0), γ(0))) -1 2 . Let K be any compact subset of T M {0} and let l(K) denote l(K) := inf X∈K {g(X, X)}.

With the notations from Section 3, for every X ∈ I D ∩ K we have L D (X) ≤ (εl(K)) -1 2 , so D is tame by Proposition 3.6.

The analytically tame condition ( 6) is clearly open in the C 1 topology defined by the metric g on the space of Weyl structures. Therefore, Theorem 4.1 applies to open subsets of the space of closed Weyl structures. 5.2. Holonomy issues. An exact Weyl structure on a conformal manifold is just the Levi-Civita connection of some metric in the conformal class. The possible restricted holonomy groups of exact Weyl structures are thus given by the Berger-Simons theorem ( [START_REF] Besse | Einstein manifolds[END_REF], p. 300). The analogous question for non-closed Weyl structures can be answered from [START_REF] Merkulov | Classification of irreducible holonomies of torsion-free affine connections[END_REF] in the irreducible case and was studied in [START_REF] Belgun | Weyl-parallel forms and conformal products[END_REF] in the reducible case. It thus remains to understand the case of closed, non-exact Weyl structures. The next result gives a complete list in the compact case, under the assumption that the connection is tame.

Proposition 5.8. The restricted holonomy group of a closed, non-exact, tame Weyl structure D on a compact n-dimensional conformal manifold (M, c) is one of the following: SO(n), U(n/2), SU(n/2), Sp(n/4), G 2 (for n = 7), Spin [START_REF] Gauduchon | La 1-forme de torsion d'une variété hermitienne compacte[END_REF] (for n = 8), 0.

Conversely, each of the groups listed above can be realized as the restricted holonomy of a closed, non-exact Weyl structure on a compact conformal manifold.

Proof. Since D is locally the Levi-Civita connection of metrics in the conformal class c, the Berger-Simons theorem applies. Assume first that D is locally symmetric. The metric g 0 on the minimal Riemannian cover M 0 of (M, c, D) is then locally symmetric. Every nontrivial homothety f satisfies f * g 0 = ρ(f ) 2 g 0 and preserves the Riemannian curvature tensor R 0 . In particular f * (|R 0 | 2 ) = ρ(f ) -4 |R 0 | 2 . On the other hand, R 0 being parallel with respect to the Levi-Civita connection of g 0 , |R 0 | 2 is constant on M 0 . Since ρ(f ) = 1, this shows that (M 0 , g 0 ) is flat, so Hol 0 (D) = 0.

Assuming from now on that Hol 0 (D) = 0, D is irreducible by Theorem 4.1, so Hol 0 (D) is in the Berger list [START_REF] Besse | Einstein manifolds[END_REF], p. 301. It remains to show that Sp(k) • Sp(1) can not be realized as the restricted holonomy group of a closed, non-exact Weyl structure. The argument is similar to the one used above. If Hol 0 (D) = Sp(k) • Sp(1) then the minimal cover (M 0 , g 0 ) is quaternion-Kähler, therefore Einstein with non-zero Ricci tensor Ric = λg 0 [START_REF] Besse | Einstein manifolds[END_REF]. Since the homotheties that act on M 0 preserve the Levi-Civita connection of g 0 , they also preserve the Ricci tensor. We infer that every homothety has to be an isometry, which contradicts the fact that D is not exact.

Conversely, we will show that every group in the above list can be realized as the holonomy of a closed, non-exact Weyl structure on a compact manifold. In fact, the examples that we provide are all quotients of cone constructions by a non-trivial homothety [START_REF] Bär | Real Killing spinors and holonomy[END_REF]. Indeed, let n > 2 (the case n = 2 is trivial) and (N n-1 , h) be a compact Riemannian manifold of a certain weak holonomy type (that we precise in each case), and let M := S 1 × N with the product metric g. The conformal manifold (M, c) (where c is the conformal class of g) can equally be seen as the quotient of the cone M 0 := R * + × N, g 0 = dt 2 + t 2 h by the infinite cyclic group generated by η(t, x) := (kt, x), for 0 < k < 1. The cone-like manifold (M 0 , g 0 ) is the minimal Riemannian cover of the Weyl manifold (M, c, D), where D is the projection to M of the Levi-Civita connection of the cone (M 0 , g 0 ).

The remaining part of the proof is based on well-known results in Riemannian geometry (see [START_REF] Bär | Real Killing spinors and holonomy[END_REF] for details):

(1) If (N 2m-1 , h) is Sasakian and not Einstein, then Hol 0 (M, D) = U(m) [START_REF] Vaisman | Locally conformal Kähler manifolds with parallel Lee form[END_REF];

(2) If (N 2m-1 , h) is Sasaki-Einstein and neither 3-Sasakian, nor locally isometric to the unit sphere, then Hol 0 (M, D) = SU(m); (3) If (N 4k-1 , h) is 3-Sasakian but not locally isometric to the unit sphere, then Hol 0 (M, D) = Sp(k);

(4) If (N 6 , h) is strictly nearly Kähler [START_REF] Gray | The structure of nearly Kähler manifolds[END_REF], has scalar curvature equal to 30, and is not locally isometric to S 6 , then Hol 0 (M, D) = G 2 ; (5) If (N 7 , h) has a proper nearly parallel G 2 -structure (i.e. the space of Killing spinors with Killing constant 1/2 is one-dimensional, [START_REF] Th | On nearly-parallel G 2 -structures[END_REF]), then Hol 0 (M, D) = Spin( 7). ( 6) If (N n-1 , h) is the unit sphere S n-1 , then Hol 0 (M, D) = 0.

Note that the case Hol 0 (M, D) = U(m) is well-known in the literature and corresponds to locally conformally Kähler (l.c.K.) manifolds. The l.c.K. structure constructed above on S 1 ×N for every Sasakian manifold N has the following special property: There exists a metric g in the conformal class such that the Lee form of D with respect to g is ∇ g -parallel [START_REF] Vaisman | Locally conformal Kähler manifolds with parallel Lee form[END_REF]. This special kind of l.c.K. metric is called Vaisman metric and it is known that not every l.c.K. structure contains such a metric in the conformal class, not even for a deformation of the l.c.K. conformal class (see [START_REF] Lebrun | Anti-self-dual Hermitian metrics on blown-up Hopf surfaces[END_REF], [START_REF] Tricceri | Some examples of locally conformal Kähler manifolds[END_REF] for examples of l.c.K. manifolds which can not be conformally Vaisman for topological reasons, having non-zero Euler characteristic, and also [START_REF] Belgun | On the structure of non-Kähler complex surfaces[END_REF] for a classification of Vaisman structures on compact 4-manifolds).

For the other holonomy groups in the above list we have the following structure result (note that the tame assumption is no longer required): Proposition 5.9. Let (M, c, D) be a compact Weyl manifold of dimension n > 2, such that D is a closed Weyl structure whose restricted holonomy is one of the following subgroups of SO(n): SU(n/4), Sp(n/4), G 2 ⊂ SO(7), Spin(7) ⊂ SO(8) or 0 ⊂ SO(n) (cases ( 2)- [START_REF] Gallot | Équations différentielles caractéristiques de la sphère[END_REF]

in the above list). Then the following hold:

(1) There exists a compact Riemannian manifold (N, g N ) satisfying one of the conditions ( 2)-( 6) above, such that the universal cover M of M, together with the metric g 0 whose Levi-Civita connection is the pull-back of D, is a metric cone over (N, g N ), i.e. M = R * + × N, and g 0 = dt 2 + t 2 g N . (2) The minimal Riemannian cover of (M, c, D) is the metric cone over a finite quotient of (N, g N ).

(3) The manifold M, endowed with its Gauduchon metric, is the mapping torus of an isometry of this finite quotient of (N, g N ).

Proof. Let g ∈ c denote the Gauduchon metric of D on M (which is determined up to a multiplicative constant by the fact that the Lee form of D with respect to g is δ g -co-closed, see [START_REF] Gauduchon | La 1-forme de torsion d'une variété hermitienne compacte[END_REF]), as well as its pull-back to the universal cover M of M. We denote by g 0 the metric on M having D as Levi-Civita covariant derivative. In all five cases (2)-( 6), the metric g 0 is Ricci-flat, so D is an Einstein-Weyl structure. This also holds on the compact manifold M, therefore Theorem 3 in [START_REF] Gauduchon | Structures de Weyl-Einstein, espaces de twisteurs et variétés de type S 1 × S 3[END_REF] implies that the Lee form of D with respect to g is parallel. The same is true on the complete, simply connected manifold ( M , g), which is therefore a Riemannian product (R, ds 2 ) × (N, g N ).

The Lee form of D with respect to g on M is just ds, so g 0 = e 2s g, i.e. g 0 = dt 2 + t 2 g N after a coordinate change t := e s . This means that ( M , g 0 ) is the metric cone over (N, g N ). It is well-known, see for example [START_REF] Bär | Real Killing spinors and holonomy[END_REF], that if the holonomy of the metric cone of (N, g N ) is one of the five groups above, then (N, g N ) is Einstein with positive scalar curvature. This, together with the fact that N is closed in M , (and thus complete), implies that N has to be compact.

Let f ∈ π 1 (M) be any deck transformation, thus acting isometrically on ( M , g). Since f is affine with respect to D, it has to preserve the Lee form of D with respect to g, i.e. f * (ds) = ds, and therefore it preserves its g-dual ∂/∂s. This means that f commutes with the flow of ∂/∂s on M , so it is induced by an isometry, also denoted by f , of (N, g N ): f (s, x) = (s + ln(ρ(f )), f (x)) (recall that ρ(f ) is the homothety constant of f with respect to g 0 : f * g 0 = ρ(f ) 2 g 0 ). It follows that the group I ⊂ π 1 (M) of deck transformations preserving g 0 induces a group of isometries I N acting freely on (N, g N ), so the minimal Riemannian cover (M 0 , g 0 ) of (M, c, D) is the metric cone over (N, g N )/I N .

Finally, the compactness of N implies that the deck transformation group Γ = π 1 (M)/I of the covering M 0 → M is discrete, hence isomorphic to Z, showing that (M, g) is the mapping torus of an isometry of (N, g N )/I N .

As a consequence, χ(M) = 0 and the fundamental group of M is a finite extension of Z. Note that if dim M = 2 and D is flat, its minimal covering may be C * or C. In both cases π 1 (M) is (a finite extension of) Z 2 .

Remark 5.10. The Berger-Simons theorem, along with the de Rham decomposition theorem, completely classify the restricted holonomy groups of torsion-free connections with bounded full holonomy group (as a subset of GL(n, R) ⊂ R n 2 ). On the other hand, a closed, non-exact Weyl structure is just a torsion-free connection whose restricted holonomy group is compact, but its full holonomy group is not bounded. Theorem 4.1 and the results in this section can thus be interpreted as an holonomy classification for this kind of connections (under the tame assumption). 5.3. An example of cone-like manifold which is not tame. Let C 0 be the following rotation cone in R 3 :

C 0 := {(x, y, z) | z = x 2 + y 2 }.

The set C 0 := C 0 {0} is a smooth Riemannian submanifold of R 3 and its metric completion is C 0 . The homothety X → 2X in R 3 defines by restriction a homothety f of C 0 , which generates a group of homotheties Γ := {f n | n ∈ Z} acting freely and properly discontinuously on C 0 . The quotient space is a topological torus T 2 . The Riemannian metric on C 0 defines by projection a conformal structure on T 2 , and its Levi-Civita connection projects to a closed, non-exact Weyl structure on T 2 .

We are going to apply some surgery and smoothening to get by similar methods a closed, non-exact Weyl structure on a surface of genus 2.

To do that, consider the domain B 0 := C 0 ∩ {1 < z < 2}, remove the two topological discs obtained as intersection of B 0 with the full cylinder Z 0 := {(x, y, z) ∈ R 3 | y 2 + (z -3/2) 2 ≤ 1/16}, connect the borders of the two removed discs by the part of the boundary of Z 0 that lies inside the cone C 0 , then smoothen it up to get a new surface B ⊂ R 3 such that:

(1) Only the part of B 0 inside the (larger) cylinder Z := {(x, y, z) ∈ R 3 | y 2 + (z -3/2) 2 ≤ 1/8} has been changed (in particular there are neighbourhoods of the two boundary circles of B 0 that are unchanged, so the gluing with the remaining part of C 0 can be done smoothly);

(2) The symmetries S x , S y : R 3 → R 3 , S x (x, y, z) := (-x, y, z), S y (x, y, z) := (x, -y, z) still act as isometries of B.

The union

N 0 := n∈Z f n B
is then a non-closed (hence incomplete) smooth submanifold in R 3 which can be completed as a metric space by adding the origin to it. Let g 0 denote the induced Riemannian metric from R 3 . The group Γ acts on (N 0 , g 0 ) by homotheties and the quotient space N := N 0 /Γ is a genus 2 surface (obtained by gluing together the two circles that constitute the boundary of B). The Riemannian metric g 0 and its Levi- We are going to study the geodesics of N 0 and prove the following Proposition 5.11. The Weyl structure D is not tame on (N, c).

Proof. Consider the isometries S x and S y acting on (N 0 , g 0 ), whose fixed point sets consist of unions of geodesics in N 0 :

(1) Fix(S x ) = N 0 ∩ {x = 0}, which is a union of two half lines c + , c -: (0, ∞) → N 0 , c ± (t) := (0, ±t, t) and an infinity of circles k n := {(0, y, z) | y 2 + (z -3 • 2 n-1 ) 2 = 4 n-2 }, n ∈ Z; (2) Fix(S y ) = N 0 ∩ {y = 0}, which is a union of closed curves γ n connecting f n (B) with f n+1 (B) and intersecting k n in P n := (0, 0, 7 • 2 n-2 ), and k n+1 in Q n+1 := (0, 0, 3 • 2 n-1 ).

We denote P := P n , for some positive integer n. The point P is a fixed point for both isometries S x and S y , and hence for their composition S := S x • S y . The latter induces the map X → -X on T P N 0 and associates to a point Q ∈ N 0 the geodesic reflection through P , i.e. the point Q such that, for any geodesic c Q : (-ε, a] → N 0 with c Q (a) = Q and c Q (0) = P , c Q can be defined on a symmetric interval [-a, a], and Q = c Q (-a).

On the other hand, there exists a geodesic γ : (-ε, T ) → N 0 such that γ(0) = P and γ(t) tends to ω = (0, 0, 0) as t tends to T . The remark above implies that the geodesic is actually defined on (-T, T ) (and this is its maximal domain of definition), and lim t→T γ(t) = ω = lim t→-T γ(t), so both ends of the incomplete geodesic γ tend to the singularity.

For any ε > 0, the point γ(Tε) can thus be connected by at least two half-geodesics with ω, namely the two branches of γ, of lengths ε and 2Tε respectively. As ε can be chosen arbitrarily small, we see that there is no bound for the ratios of those lengths, therefore N 0 is not tame by the converse statement in Proposition 3.11. 5.4. Open problems. Several natural questions emerge from the considerations in the present paper. We list here some of the most interesting ones:

(1) Theorem 2.6 shows that if D is a closed Weyl structure on a compact conformal manifold (M, c), then the minimal Riemannian cover (M 0 , g 0 ) can be metrically completed by adding exactly one point. The metric completion of its universal covering ( M , g) is, however, not well understood: The boundary of M in its metric completion may be more complicated in general, possibly depending on the growth of the fundamental group of M. (2) One can check that the analytically tame condition forces the Lee form θ to be non-vanishing, therefore restricting the topology of M 0 to products R × N, in particular χ(M) = 0. Does the tame condition also imply a topological restriction? And if this restriction is satisfied, is the structure automatically tame? (3) Ultimately, is the tame condition necessary in the proof of Theorem 4.1?

The answer to this last question seems to be the most challenging problem in the holonomy theory of Weyl structures.

Lemma 2 . 1 .

 21 The group A ≃ π 1 (M) of deck transformations of the covering M → M consists of homotheties of g0 .

Definition 3 . 2 .

 32 An affine connection D on a manifold M is called weakly tame if the set C D of non-zero vectors generating a complete half-geodesic is open in T M {0}.

Proposition 3 . 6 .

 36 An affine connection D on M is tame if and only if L D is bounded on any intersection of I D with a compact set in T M {0}.

Theorem 4 . 1 .

 41 If the restricted holonomy representation of a closed, non-exact, tame Weyl structure D on a compact conformal manifold (M, c) is reducible, then D is flat.

Figure 3 .

 3 Figure 3. Idea of the proof of Lemma 4.3 (ii).

Figure 4 .

 4 Figure 4. Idea of the proof of Proposition 4.5.

) Lemma 4 . 9 .

 49 For every x ∈ M 0 and positive real number r, the open ball B x (r) contains the open ball Bx (r), where r = r k 2 (δ(x)+r) .

  Civita connection define, by projection, a conformal structure c, and a closed, non-exact Weyl structure D on N.

Figure 5 .

 5 Figure 5. Construction and properties of N 0 .

This work was partially supported by the French-German cooperation project Procope no. 17825PG.


est destinée au dépôt

Investigation of the organic anion production by helium ion irradiation of calcite suspensions in water

Context

This study deals with the deep geological storage of long-lived radioactive wastes. In particular, among the options available, the use of concrete has been extensively studied. However, the presence of radionuclides in the waste container will however induce radiolysis phenomena on the concrete and the cement matrix. The carbonatation of portlandite Ca(OH) 2 in calcite CaCO 3 is a natural phenomenon occuring in every concrete depending on the storage conditions, and especially in the nuclear waste storage conditions Where does acetate come from?

Are formate and acetate directly responsible for oxalate formation?

Irradiation of organic anions solutions in water

  [1]. The radiation chemical behavior of calcite is not well documented yet[2]. It is thus difficult to predict the evolution of chemical properties of concrete under irradiation, if the latter was carbonated. « Healthy » cement : Portlandite Ca(OH) 2 Carbonatation Calcite CaCO 3 Impact of ionizing radiation on calcite? • Helium ion irradiation of calcite samples • ARRONAX cyclotron (Nantes) Energy 60,7 MeV • Focus on the organic ions produced by the irradiation Irradiation of calcite suspensions in water [1] M. Thiery, , Ecole nationale des ponts et chaussées (2005) [2] G. Albarrán Radiat. Phys. Chem. (1994) 43, 307 [3] T.E. Eriksen, J. Nucl. Mater. (2012) 375 (3carbonate solutions Calcite acts as a carbonate ion supply, thus providing calcium accumulation in the medium Questions:

SUBATECH, UMR 6457, IMT Atlantique, CNRS/IN2P3, Université de Nantes, 4 rue Alfred Kastler BP

20722, 44307 Nantes Cedex 3 France 2 Cyclotron ARRONAX, 1 rue ARRONAX, CS 10112, 44817 Saint Herblain cedex, France