On the topology of closed manifolds with quasi-constant sectional curvature

— We prove that closed manifolds admitting a generic metric whose sectional curvature is locally quasi-constant are graphs of space forms. In the more general setting of QC spaces where sets of isotropic points are arbitrary, under suitable positivity assumption and for torsion-free fundamental groups, they are still diffeomorphic to connected sums of spherical bundles over the circle. Résumé (Sur la topologie des variété fermées à courbure sectionnelle quasi-constante) Nous montrons que les variétés fermées admettant une métrique générique dont la courbure sectionnelle est localement quasi-constante sont des sommes graphées de variétés de courbure constante. Ensuite nous étendons ce résultat au cas des espaces QC dont l’ensemble des points isotropes pourrait être arbitraire en démontrant que, sous une condition de positivité et lorsque leurs groupes fondamentaux sont sans torsion, ils sont difféomorphes à des sommes connexes de fibrés en sphères sur le cercle.


Introduction
The classification of (locally) conformally flat closed manifolds seems out of reach in full generality since M. Kapovich proved (see [34]) that an arbitrary finitely presented group G is a subgroup of a free amalgamated product of the form G * H which is a fundamental group of a conformally flat closed manifold of dimension (at least) 4.
It is proved in [6] that the sectional curvature of a globally 1-QC manifold is constant along any cone of 2-planes making a constant angle in [0, π/2] with ξ, in every point.
Recall that a line field is a section of the projectivized tangent bundle. A line field ξ on M determines a real line bundle, still denoted ξ over M . Its first Stiefel-Whitney class w 1 (ξ) ∈ H 1 (M, Z/2Z) vanishes if and only if ξ has a section, namely a non-zero vector field ξ defining the same line field. When this is the case, the line field will be called orientable and such a vector field ξ will be called a lift of the line field. When ξ is non-orientable, its pull-back to the 2-fold cover M of M associated to the class w 1 (ξ) is orientable and defines a vector field ξ on M . We call ξ the distinguished vector field associated to the globally 1-QC manifold with distinguished line field ξ.
Throughout this paper we will use the term cylinder in its topological sense, namely, a cylinder is a manifold diffeomorphic to the product of a compact manifold with an interval (which might be open or closed). More generally, a twisted cylinder is an interval bundle over a compact manifold, which might possibly be a non-trivial bundle.
The topology of globally 1-QC manifolds is rather simple, as we will see that they must be 2-fold covered by cylinders. Further, the geometry of globally 1-QC manifolds with orientable line field was completely described under an additional condition (namely, that the distinguished field be geodesic) in [22, §6]: these are precisely the warped product Riemannian manifolds between some closed space form and an interval which are called sub-projective spaces. Notice that the vector field ξ is (locally) conformally Killing if and only if it is Killing and if and only if it is geodesic.
In order to enlarge the number of possible topologies arising in the definition above we slightly weaken it as follows. (2) 1-QC if any non-isotropic point for the sectional curvature function has an open neighborhood which is locally 1-QC.
Note that line fields do not necessarily define a global line field on the manifold M . Observe that a locally 1-QC manifold is a 1-QC manifold.
Recall that a point of a Riemannian manifold is called isotropic if all its tangent 2-planes have the same sectional curvature. Denote by C the set of isotropic points of M . In the case of a globally 1-QC manifold the sectional curvatures N and H of 2-planes containing and respectively orthogonal to the distinguished line field ξ will be called vertical and horizontal curvature functions, respectively. It will be shown in the next section that both H and N extend smoothly from the globally 1-QC neighborhoods to all of M , when the manifold M is 1-QC. Moreover, the set of non-isotropic points carries an unique distinguished line field extending the distinguished line fields associated to various locally 1-QC neighborhoods.
For a 1-QC manifold M , the set C of its isotropic points then consists of those points p ∈ M satisfying the equation: Our main issue in this paper is to give some insights on the topology of C and M C. Results of foliation theory give the description of M C as twisted cylinders. Further, we will prove that M C and int(C) can be approximated by manifolds with boundary when M is locally 1-QC. A graph of space forms is a closed manifold M which can be obtained from a finite set of compact space forms with totally umbilical boundary components by gluing isometrically interval bundles (twisted cylinders) along boundary components. The pieces will be accordingly called vertex manifolds and edge manifolds or tubes.
Note that the induced metric on the graph of space forms is not necessarily smooth and its diffeomorphism type depends on additional choices, like the gluing maps. In particular, fiber bundles over the circle whose fibers are closed space forms are such graphs of space forms.
The structure of conformally flat closed manifolds whose universal covering have a conformal vector field was described by Goldman and Kamishima (see [24]). This includes the case of globally 1-QC manifolds with geodesic distinguished field ξ. Motivated by this, our first result is a general description of the topology of the locally 1-QC spaces, under a mild genericity condition, as follows: Theorem 1.6. -Let M n be a closed locally 1-QC n-manifold, n 3, whose metric is generic and assumed conformally flat when n = 3. Assume that the rank of H 1 (M C; Q) is finite. Then M is diffeomorphic to a graph of space forms. The curvature leaves function λ is defined at non-isotropic points (see also Section 2.5) by: Our second result aims to extend Theorem 1.6 to the case of 1-QC manifolds under additional hypotheses.
Remark 1.12. -Rational Pontryagin forms, and hence classes, vanish on conformally flat manifolds, see [15,37]. Conversely, it seems unknown whether conformally flat manifolds admit conformally flat metrics for which the curvature tensor in a suitable basis has all its components with 3 and 4 distinct indices vanishing.
Throughout this paper (local) 1-QC manifold or space will mean a manifold which admits a Riemannian (local) 1-QC metric.
The structure of this paper is as follows. In Section 2 we collect a number of general results concerning 1-QC manifolds for further use. We show that the horizontal and vertical curvature functions extend smoothly to the entire manifold first in the locally 1-QC case (Lemma 2.2), then using the higher Weitzenböck curvatures in the general 1-QC case (Lemma 2.7). In dimensions n 4, 1-QC manifolds are conformally flat, while this has to be added to our assumptions when n = 3 (see Lemma 2.9). The level hypersurfaces of the horizontal curvature define a partial foliation with complete leaves which are totally umbilical submanifolds of intrinsic constant curvature λ. We show that for locally 1-QC manifolds λ extends to the closure of the open set of non-isotropic points (Lemma 2.11). Eventually, we show that the 1-QC manifolds of dimension n 4 are precisely those Riemannian manifolds whose universal coverings have codimensionone isometric immersions into the hyperbolic space, which are equivariant with respect to some holonomy homomorphism (Propositions 2.14-2.18). Section 3 is devoted to the proof of Theorem 1.6. We start by describing the topology of globally 1-QC manifolds with non-empty set of isotropic points: they are twisted cylinders over compact space space forms (Proposition 3.1), since the curvature leaves are compact (Proposition 3.3). This is a consequence of deep results of Reckziegel ([47,48]) concerning the completeness of curvature leaves and Haefliger's classical theory of foliations with compact leaves ( [28]). In Section 3.8 we collect a few results to be used later about the geometry of the curvature leaves, by showing that their second fundamental form, injectivity radius and (local) normal injectivity are uniformly bounded. The existence of a well-defined distinguished vector field ξ in a neighborhood of an isotropic point shows that curvature leaves have a controlled behavior when they approach the isotropic point, by limiting to some branched hypersurface (Proposition 3.29). We further need to analyze the degeneracies occurring in the closure of these components by using the branched hypersurfaces which arise as limit leaves (Proposition 3.34). We show that we can choose finitely many limit leaves such that the open isotropic components can be compactified to manifolds with boundary (Propositions 3.40 and 3.42). By gluing isometrically some cylinders endowed with a warped product metric along the boundary we construct space forms with umbilical boundary.
In Section 4 we are concerned with the proof of Theorem 1.8. The distinguished line field cannot be extended to isotropic points as in the previous section but the curvature leaves are now constrained to be diffeomorphic to spheres, as they have positive curvature. Every connected component of a saturated neighborhood of the set of isotropic points is a manifold whose boundary components are spheres (Proposition 4.9). Group theory arguments involving Grushko's theorem prove that there are only finitely many spheres which do not bound a ball (Proposition 4.5). The intrinsic curvature of a spherical leaf provides control on the size of a spherical cap in the hyperbolic space which bounds that leaf (Lemmas 4. 8 and 4.18). We then prove that we can realize the capping off the boundary spheres in the Riemannian setting and the closed manifold obtained this way is diffeomorphic to a space form, according to a theorem of Nikolaev ([44]) (see Proposition 4.17).
In Section 5 we complete the description of isometric immersions of 1-QC conformally flat manifolds with the case of dimension n = 3 (Proposition 5.1). Further, we note that a 1-QC metric admits an extension to a hyperbolic metric on the product M × [0, ∞) (Proposition 5.3).
Then, by Lemma 2.1, every point p ∈ C UiUj is isotropic for the sectional curvature. This implies that H Ui (p) = N Ui (p) = H Uj (p) = N Uj (p).
Therefore, we have a finite collection of smooth functions H Ui and N Ui whose restrictions to intersections U i ∩U j coincide. Thus they define smooth functions on M .
Recall that C denotes the closed subset of M of isotropic points for the sectional curvature. We take a refined open cover {V j } such that every V j is contained into some U i . If p ∈ (V i ∩ V j ) C, then ξ Vi | p = ξ Vj | p , by Lemma 2.1. This closed condition is therefore valid for any point p ∈ (V i ∩ V j ) C. It follows that the line fields ξ Vi agree at any point p ∈ M C. They define therefore a distinguished line field on M C.
When the line field is orientable, it can be extended to an open set Ω containing M C, by a classical partition of unity argument. Otherwise, we construct an extension on the 2-fold cover of an open neighborhood of M C associated to the first Stiefel-Whitney class of ξ and then project it down.
Let ∆ be the Hodge-de Rham Laplacian ∆ = dd * + d * d and ∇ * ∇ be the Bochner Laplacian of the Riemannian manifold M . Recall that ∇ : is the covariant derivative induced by the Levi-Civita connection on the space of m-forms Ω m (M ) and ∇ * its adjoint, namely for any vector fields X j and an orthonormal local basis {e i }.
If ω is a m-form, then the Weitzenböck formula states that: Here, R m is the m-th Weitzenböck curvature operator, namely a degree zero pseudodifferential operator determined by the Riemann tensor: The Riemannian connection induces a metric connection in the bundle Ω m (M ) and its curvature R used above is a section of Ω 2 (M ) ⊗ Ω m (M ). Let now consider the real function defined on M by the formula: where λ 1 λ 2 · · · λ n denote the eigenvalues of the Schouten operator S defined as in terms of the Ricci tensor Ric and the scalar curvature R. Assume that the Weyl tensor of M vanishes so that the Riemann tensor satisfies the following identity (see [42]): It follows by direct calculation (see [27,App.]) that R m ω = G m ω. In particular, we obtain the following: Lemma 2.4. -Let M be a Riemannian n-manifold, n 3, whose Weyl tensor vanishes. Then the function G m : M → R is smooth.

2.3.
Isotropic points of 1-QC manifolds. -In this section M is a 1-QC manifold and C denotes the closed set of its isotropic points. The first observation is: -If M is a compact 1-QC then M C is globally 1-QC. In particular, H and N are smooth real functions defined on M C.
Proof. -According to the definition, M C is locally 1-QC, hence it admits a covering by open sets U i ⊂ M C which are globally 1-QC with respect to line fields ξ Ui . However, points p ∈ U i ∩ U j where the corresponding line fields do not agree should be isotropic, following Lemma 2.1 and the proof of Lemma 2.2, which contradicts p ∈ M C. Thus the distinguished line field is well-defined and M C is a globally 1-QC manifold. Proof. -This is known for points of M C (see [5,22]) and is classical for isotropic points.
Let ξ be a locally defined unit vector field which determines the line field ξ. Moreover, we denote by the same letter ξ an arbitrary extension (not necessarily continuous) to all of M . Proof. -Let R denote the scalar curvature and Ric the Ricci curvature. We set then: The formulas for the Ricci tensor from [5,6] imply that at all points p ∈ M C. Note that g(X, ξ)g(Y, ξ) is well-defined on M C and independent on the choice of the vector field ξ. Nevertheless, this formula makes sense and holds at points p ∈ C, as well.
The Ricci operator has eigenvalues (n − 2)H + N and (n − 1)N with multiplicities n − 1 and 1, respectively at points of M C and (n − 2)H + N = (n − 1)N with multiplicity n at points of C.
The Schouten tensor S is from above: The eigenvalues of the Schouten operator are therefore H/2 and N − H/2, with multiplicities n − 1 and 1, respectively at points of M C and H/2 = N − H/2 with multiplicity n at points of C. According to Lemmas 2.4 and 2.6 the function G m is smooth on M . An explicit calculation shows that: If n = 4, since R, G 1 and G 2 are smooth, we derive that min(H, N ) and then N and H are smooth functions on M . When n = 4, G 1 + G 3 = 2H + 4N and R/3 = 2N + 2H are smooth, so H, N and min(H, N ) are smooth. -Let M be a compact 1-QC manifold. We define the horizontal distribution D to be the orthogonal complement to the distinguished line field ξ. By Lemmas 2.5 and 2.3, the distribution D is well-defined on M C. One important ingredient used in this paper is the following easy observation (see also [5,22]): (1) 1-QC manifolds of dimension n 4 are conformally flat.
(2) 1-QC manifolds of dimension n = 3 are conformally flat if and only if the horizontal distribution D is completely integrable on M C.
By the Weyl-Schouten theorem (see [42,§C,Th. 9]) the 3-manifold M is conformally flat iff the so-called Cotton-Schouten tensor vanishes, namely if: Given p ∈ M C, let U be a disk neighborhood of p within M C. Define the 1-form η on U by η(X) = g( ξ| U , X), where ξ| U is a vector field generating ξ| U . Then the horizontal distribution D is the kernel of η on U . The vanishing of the Cotton-Schouten tensor on U is equivalent to: Using all possibilities for choosing X, Y, Z among the elements of a basis of the horizontal distribution D or ξ| U it follows that this is equivalent to the system of the following three equations: The first two equations are always verified by using the method of [6,22] in all dimensions n 3. The third one is equivalent to the fact that the horizontal distribution D is completely integrable on U . Thus D is integrable if M is conformally flat. Conversely, if D is integrable, the Cotton-Schouten tensor vanishes on M C, and hence on M C. On the other hand M M C coincides with the set int(C) of interior points of C, namely those points of C having an open neighborhood in M which is contained in C. By Schur's theorem H = N are constant in any connected component of int(C) and in particular the Cotton-Schouten tensor vanishes. This proves that M is conformally flat.
Remark 2.10. -One cannot dispose of the integrability condition for D when n = 3. In fact, any closed 3-dimensional nilmanifold is covered by the Heisenberg group, endowed with a left invariant Riemannian metric. Then the tangent space at each point is identified to the nilpotent Lie algebra generated by X, Y, ξ with relations [X, Y ] = ξ, [X, ξ] = [Y, ξ] = 0. The horizontal distribution corresponds to the subspace generated by X, Y and is well-known to be non-integrable. However, the left invariant Riemannian metric on the Heisenberg group is globally 1-QC, as we can compute H = −1/4 and N = 3/4. This furnishes the typical example of a closed 1-QC 3-manifold which is not conformally flat. In fact closed nilmanifolds are not conformally flat (see [23]) unless they are finitely covered by a sphere, a torus or S 1 × S 2 .
2.5. The leaf curvature function λ. -Let M be a 1-QC manifold, which is assumed to be conformally flat, when n = 3. Then the distribution D is integrable and hence defines a foliation of M C. The leaves of the foliation will be called curvature leaves. According to [6,22] every curvature leaf is a totally umbilical submanifold of M of intrinsic constant curvature equal to: Observe that the right hand side above is independent on the choice of a local distinguished unit vector field ξ which lifts ξ.
Before we proceed, recall that a real function f defined on a compact subset K of a manifold M is called smooth if for every point p ∈ K there exists an open neighborhood U ⊂ M containing p and a real smooth function Proof. -For every point p of M we choose an open neighborhood V ⊂ M which is contractible so that the distinguished line field ξ V is defined and can be lifted to a vector field ξ V . Let X V be a unit vector field on V orthogonal to ξ V . According to [6, (3.11)] and [22, (3.5)] we have: The function (g(∇ X V ξ V , X V )) 2 is smooth and well-defined on V , namely independent on the choice of the lift ξ V . We set therefore where X V is an arbitrary smooth unit vector field on V orthogonal to ξ V . Then λ V is a smooth function on V .
Therefore the collection λ Ui defines a smooth extension λ : M C → R.
2.6. Isometric immersions of 1-QC manifolds. -One important tool which will be used throughout this paper is the codimension-one immersability of simply connected 1-QC manifolds. Let H m κ denote the m-dimensional hyperbolic space of constant sectional curvature −κ, when κ > 0 and the Euclidean space when κ = 0, respectively. For a 1-QC manifold M n we denote The main result of this section is: 14. -Assume that M n , n 3, admits a 1-QC metric which is conformally flat when n = 3. Then there exists an equivariant isometric immersion, namely an isometric immersion f : M → H n+1 κ , and a group homomorphism where the action on the right side is the action by isometries of Iso(H n+1 κ ) on H n+1 κ while the left side action is by deck transformations on the universal covering.
Proof. -The universal covering M is a 1-QC manifold with the induced metric by the covering π : M → M .
Every p ∈ M C has an open contractible neighborhood such that the line field ξ can be lifted to a vector field ξ U . We derive a smooth tensor h U (X, Y ) = g( ξ U , X)g( ξ U , Y ) on U . If p ∈ U ∩ V then ξ U | p = ± ξ V | p and thus h U (X, Y )| p = h V (X, Y )| p , for any p ∈ U ∩ V . Thus the collection h U , where {U } is a covering of M C defines a smooth tensor field h on M C.
Define now the following smooth symmetric (1,1) tensor field on M pointwise, by means of the formula: and also denote by h the pull-back to M . One verifies immediately the Gauss and Codazzi-Mainardi equations associated to the tensor h, namely: This follows from the second Bianchi identity of the curvature tensor when n 4, and from the vanishing of the Cotton-Schouten tensor in dimension 3. The second equality is treated as in the proof of Lemma 2.9 by noticing that only the last condition discussed there has to be verified, the other ones being automatic.
The second fundamental form h on M is invariant by the action of the deck group, by construction. This means that there exist open neighborhoods of x and γ · x which are isometric. Let ρ(γ) x be the global isometry sending one neighborhood onto the other. Now ρ(γ) x is locally constant, as a function of x, and hence it is independent on x. Furthermore ρ(γ) is uniquely defined by the requirement to send n+1 points x i in general position in H n+1 κ , which belong to a small open neighborhood of x into their respective images γ · x i . Therefore, if the image of the immersion is not contained in a hyperbolic hyperplane then ρ(γ) is unique and this also implies that ρ is a group homomorphism. In the remaining case M is hyperbolic and the result follows again. Contrasting with the 2-dimensional case, but in accordance with Schur's theorem, the space of such isometric immersion is discrete. The first and simplest case is: Proof. -If h is the second fundamental form of such an immersion, then linear algebra computations show that the Gauss equation already determine h up to a sign. A hyperbolic isometry reversing orientation will change the sign of h. Now an isometric immersion is uniquely determined by its second fundamental form. , and a group homomorphism ρ : Proof. -According to Cartan (see [10] and [42,Th. 4]) at each point p ∈ M there exists a principal curvature µ (i.e., an eigenvalue of the second fundamental form) of the immersed manifold of multiplicity at least n − 1. Let N be the open set of non-umbilical points, namely those for which µ has multiplicity exactly n − 1. The distribution D is the eigenspace for the shape operator corresponding to the multiple eigenvalue. From Gauss' equations we obtain that N is globally 1-QC, with distinguished line field given by the normal line field to D. The complement of N is totally umbilical in H n+1 κ and hence it has constant curvature. Since the immersion is equivariant these properties descend to M .
Remark 2.19. -Note that the statement of Proposition 2.18 is not true when n = 3 as Cartan's theorem does not extend to n = 3 (see [42]).
3. The topology of locally 1-QC manifolds 3.1. Globally 1-QC manifolds with orientable line field. -The purpose of this section is to prove that the globally 1-QC manifolds have a rather simple topological structure. Throughout this section M is a conformally flat 1-QC manifold of dimension n 3. Let E be a connected component of M C. By Lemma 2.5 the open set M C is globally 1-QC. In order to have a uniform approach we denote by E the 2-fold cover determined by the first Stiefel-Whitney class of ξ| E when ξ| E is non-orientable, and E itself otherwise.
Moreover, M C is endowed with a smooth foliation D, which is orthogonal to the line field ξ. Following [48] the leaves of the foliation D will be called curvature leaves, as the principal curvatures are constant along them. Proof. -With respect to the pull-back metric to the cover the slices P × {a} of E have constant curvature λ. But a closed manifold P cannot support constant curvature metrics with different signs. In fact π 1 ( P ) is either hyperbolic and not virtually cyclic, virtually abelian or finite, according to whether λ is negative, null or positive.
The main step is to establish the following: Proof of Proposition 3.1 assuming Proposition 3.3. -The strong form of the global Reeb stability theorem states that a codimension-one transversely orientable foliation whose leaves are all compact is locally a fibration and the holonomy groups are finite (see [43,18]).
Outline of the proof of Proposition 3.3. -We use a theorem of Reckziegel to prove that the foliation has complete leaves with respect to the induced metric in Section 3.2.
Since H is constant along the curvature leaves we derive that those leaves on which the value of H is non-critical should be compact in Section 3.5. This can be improved to those leaves which can be approached by non-critical leaves by using a Theorem of Haefliger (Section 3.3) which states that the union of closed leaves is closed. It remains then to analyze the leaves for which H is constant on an open neighborhood. Such a maximal saturated open set on which H is constant admits a metric completion whose general structure was described by Dippolito (see Section 3.4). We will first show that the completion is transversely compact and further that the connected components of its boundary are compact, as they can be approached by non-critical and hence compact leaves, in Section 3.6. We derive then that the completion is a foliated product and hence all leaves are compact.

3.2.
Completeness of curvature leaves. -Let f : N → P be an isometric immersion of the Riemannian manifold N into a Riemannian manifold P with constant curvature and dimension strictly larger than that of N . Set ν(f ) for the normal bundle of f , ν * (f ) for its dual and II : T N × T N → ν(f ) for the second fundamental tensor of f . The shape operator A η : T N → T N is defined by the identity g(A η X, Y ) = g P (II(X, Y ), η), for all sections X, Y of T N and η of ν(f ), where g P and g are the metric tensors on T P and ν(f ), respectively.
A principal curvature function is a continuous section µ of ν * (f ) for which the eigenspace E (µ(p)) has dimension dim E (µ(p)) 1 for each p ∈ N , where we put Let U be an open subset of N for which dim E (µ(p)) is constant, say equal to k, for every p ∈ U .
A key ingredient of the proof of  (1) The principal curvature function µ is smooth on U .
(2) The vector spaces E (µ(p)), for p ∈ U , form a subbundle E (µ) of T N | U which is integrable.
(3) If L is a leaf of the foliation defined by the subbundle E (µ) then L is a totally umbilical submanifold of N of dimension k and f | L : L → P is a totally umbilical immersion.
(5) If k 2, N is complete and U is the subset of those points p ∈ N for which dim E (µ(p)) is minimal (U is open since this dim E (µ(p)) is upper semicontinuous), then every leaf L of the foliation of U defined by E (µ) is a complete metric space.
Consider now a conformally flat 1-QC manifold M of dimension n 3. Proposition 2.14 shows that there exists an equivariant isometric immersion f : M → H n+1 κ . Set π : M → M for the universal covering projection. Observe now that one principal curvature µ of the immersion f is of multiplicity n − 1 precisely along the open set U = π −1 (M C). Moreover, the pull-back π * D of the distribution D by π coincides with the subbundle E (µ) on U .
Reckziegel's Theorem 3.4 in the case N = M and P = H n+1 κ therefore has the following corollary.    The structure of open saturated subsets U of a foliated manifold W was described by Dippolito ([17]) in the case when W is compact and it was extended to relatively compact saturated subsets U of open manifolds in [7]. Specifically, let u U denote the completion of U with respect to the induced Riemannian metric from W . Note that, in general, u U is non-compact. Then the inclusion i : U → W induces an immersion u i : u U → W , by extending i to limit points of trajectories of a transverse foliation. Dippolito U which is tangent to D and K is non-empty, then K ∩ L is non-empty and connected.
The nucleus is not unique in general. If we can take K = ∅, then u U = L × [0, 1] is a foliated product, foliated by the leaves of the foliation diffeomorphic to L and also transversally by closed intervals.
We will need in the sequel the following result stated by Dippolito [17,Prop. 2] in the compact case and by Cantwell and Conlon [7,Lem. 4.2] in the open case. Note that the number of such boundary components is bounded by twice the number of so-called biregular charts needed to cover U , which is finite since U is compact (see [17]).  If π : M → M is the universal covering projection, then each component of the preimage π −1 (L) of L in M is complete, following Reckziegel's Lemma 3.5. This implies that the leaf L itself should be geodesically complete, as otherwise we could adjoin additional limit points to π −1 (L) contradicting its completeness. Thus L is complete with respect to its intrinsic metric.
As a word of warning the projection π is not a closed map, if the covering has infinite degree. In particular, L is not necessarily a complete subspace of M and in particular it might not be a closed subset of M . If has an induced Riemannian metric from M whose restriction to its open subset L coincides with the intrinsic metric of L. Therefore L is a complete subspace of the compact space H −1 (H(L)) and hence it has also to be closed. This implies that L must be a connected component of the level hypersurface H −1 (H(L)) and hence compact.
Note that, as D is an integrable distribution on M C, all curvature leaves, in particular critical leaves, are nonsingular.
Suppose now that H is everywhere non-constant, namely that there is no open subset U ⊂ M C for which the restriction of H to U is constant. We want to apply Haefliger's result to the foliation by curvature leaves of the open manifold M C. Assume that the set of non-compact leaves is non-empty. By Lemma 3.6 the union U of non-compact leaves is an open subset of M C. Let p ∈ U , so that L p is a noncompact leaf and H(L p ) = c is a critical value of H. Consider a germ of an integral curve γ ⊂ U of the line field ξ. Then the restriction of H to γ is non-constant, as otherwise the union of leaves intersecting γ would contain an open set on which H is constant, contradicting the fact that H is everywhere non-constant. It follows that H(γ) contains a small interval and hence c can be approached by a sequence of regular values c i ∈ H(γ). Take points p i ∈ γ with H(p i ) = c i . Therefore the leaves L pi are non-critical leaves and hence compact leaves by above. This contradicts the fact that U is the union of non-compact leaves. This ends the proof of Proposition 3.3 for the case of everywhere non-constant H.  Proof. -Since H is constant in a neighborhood of p, we have dH| p = 0. Since p ∈ M C, we can write dH = ξ(H)η, for some locally defined vector field ξ lifting the line field ξ. Thus ∇ X ξ = αX = ( ξ(H)/(N − H))X vanishes at p, for any horizontal vector field X. In particular α(p) = 0 and λ(p) = H(p), according to the proof of Lemma 2.11. Since λ is constant on the level hypersurfaces of H it follows that λ(q) = H(q) = c 0 , for any q ∈ L p and thus the curvature leaf L p is totally geodesic.
Let E be the connected component of M C containing p, E the cover of E determined by w 1 (ξ| E ) and ξ be a vector field lift of the line field. The cover E is endowed with a pull-back metric and a completely integrable distribution still denoted g and D, respectively. We keep the notation U for the lift of U and work henceforth on E in this section. Note that E is contained in Ω and has compact closure there.
We want to construct a compact submanifold N ⊂ E containing U which is foliated by curvature leaves. Possibly enlarging U we can assume that U is the maximal open connected saturated subset of E containing p such that For every p ∈ U we consider the flow ϕ p (t) determined by ξ with initial condition ϕ(0) = p. Then ϕ p (t) is not necessarily defined on all of R, since ξ was only defined on an open neighborhood Ω of M C. Let I p 0 be the maximal interval consisting of t ∈ R for which ϕ p (t) ∈ E. Set further J p ⊂ I p for the maximal interval on which H • ϕ p is constant, namely such that 0 ∈ J p and all t ∈ J p satisfy H(ϕ p (t)) = H(p). Then J p is non-empty, because it contains the connected component of 0 in ϕ −1 p (U ). Proof.
-If I p has a finite supremum (or infimum), then ϕ p (t) reaches C in finite time.
In this case sup J p < sup I p . Otherwise H would be constant on the set U , which is the union of all leaves intersecting ϕ p (I p ) (actually a small open neighborhood of it), while U ∩ C = ∅. This would contradict the genericity of H.
If I is unbounded, say it contains [0, ∞), we have two cases: either the forward limit set ω(ϕ p ) intersects C, or else ω(ϕ p ) ⊂ E. Recall that the forward limit (or ω-limit) set of the flow ϕ p is the set of points obtained as limits of sequences ϕ p (s i ), where s i → ∞. In particular, ω(ϕ p ) is a compact set invariant by the flow.
In the first case, if sup J p = ∞, then again H is constant on the union U of leaves intersecting ϕ(I p ), while U ∩ C = ∅, contradicting the genericity of H.
Consider now the second case, when ω(ϕ p ) ⊂ E and assume that J p ⊃ [0, ∞). Recall that H| E is non-constant and thus there exist non-critical values c = H(p). Let then L q be a non-critical leaf through a point q ∈ E, with H(L q ) = c. By Lemma 3.10, L q is a compact leaf.
Further ω(ϕ p ) ∩ L q = ∅, since H(ω(ϕ p )) = H(p) and H(L q ) = c = H(p). Since both sets are compact, the distance between ω(ϕ p ) and L q is non-zero. Let then β : [0, 1] → E, with β(0) ∈ ω(ϕ p ), β(1) ∈ L q be a minimal geodesic in M realizing the distance between them. Since ω(ϕ p ) is invariant by the one-parameter flow generated by ξ the integral orbit O of β(0) under this one-parameter flow is contained in ω(ϕ p ). Standard variational arguments show that β must be orthogonal to both the orbit O and the leaf L q , namely that Write then β(0) = lim ϕ p (s i ), where s i → ∞. All leaves L ϕp(si) are super-critical (by our assumption on J) and hence α(x) = 0, for every x ∈ L ϕp(si) . Since α is smooth, we derive that α(x) = 0 for all x ∈ L β(0) , which means that L β(0) is a totally geodesic leaf.
Eventually, a minimal geodesic in M whose direction is tangent to a totally geodesic leaf must be contained in that leaf. On the other hand the second equation above shows that β is orthogonal to the leaf L q at β(1), which is a contradiction.
Note that Lemma 3.12 already shows that u U is foliated transversely by closed intervals.
Proof. -The image u i( u U ) of the completion u U is the subset of Ω obtained by completing every transverse trajectory of ξ, namely replacing every open interval ϕ p (J p ) by the corresponding closed interval (or circle) ϕ p (J p ).
In general, one can have part of the border tangent to the foliation D and part of it tangent to some trajectory of ξ (see [7, §4]). However, the second case cannot occur here because E is relatively compact within Ω, more precisely it is contained in the pull-back of E to the double cover Ω. Thus every point of δ is also the union of the corresponding curvature leaves passing through these points.
As J p is a closed proper interval within I p , the restriction H| ϕp([sup Jp,sup Jp+ε])) is not constant and hence there exists a sequence of non-critical values of the form H(ϕ p (sup J p + t i )) approaching H(p), for some decreasing t i → 0. Since non-critical leaves are compact, Haefliger's Theorem 3.6 implies that the leaves L ϕp(sup Jp) and L ϕp(inf Jp) are compact.
There are only finitely many components of ∂ u U and hence δ( u U ), by Lemma 3.9. It follows that u U is a compact manifold. Now, every pair of points p, q ∈ ∂ u U having the same image by u i, give raise to two leaves which are tangent at u i so that they should coincide. Therefore N = u i( u U ) is a codimension zero compact submanifold of E. Specifically, N is the closure of the connected component of int(H −1 (H(p)) ∩ E) containing p. Proposition 3.3 will then follow from the following: Proposition 3.14.
-The submanifold N is diffeomorphic to a cylinder foliated as a product.
Proof. -Lemmas 3.12 and 3.13 and Dippolito's description from Section 3.4, show that U has an empty nucleus, and hence u U is a foliated product by closed intervals. Here is a direct proof. Each component of ∂N has a sign determined by the fact that ξ points either inward or outward with respect to N . This is well-defined because ξ is a smooth vector field on E orthogonal to ∂N , so that if ξ points inward at some point of ∂N then it points inward at all points of its connected boundary component.
Let ∂ + N be the union of those components of ∂N for which ξ is inward pointing and ∂ − N be its complementary. Observe that N cannot be a closed manifold without boundary since it is contained in E. By the symmetry of the situation we can assume that ∂ + N is non-empty. We claim now that: Proof. -Assume the contrary. As the trajectory cannot cross again ∂ + N , necessarily in outward direction, because this would contradict the choice of ∂ + N , it will remain forever in N . Thus its forward limit set is a compact invariant subset of N . The argument used in the proof of Lemma 3.12 shows that this is impossible.
Let then denote by (x) the length of the integral trajectory ϕ x of ξ between x and its endpoint on ∂ − N . We consider then φ : Then φ is a diffeomorphism on its image. The only obstruction for the surjectivity of φ is the existence of closed trajectories of ξ. Again, the argument used in the proof of Lemma 3.12 shows that there are no closed orbits of ξ contained in int(N ). This proves Proposition 3.14.

3.7.
Another proof of Proposition 3.3. -We first recall that two points of a foliated manifold are Novikov equivalent if they belong to the same leaf or to a closed transversal. An equivalence class of points is called a Novikov component of the foliation. The following classical result is due to Novikov. An alternative way to complete the proof of Proposition 3.3, which does not use Proposition 3.14 is as follows. If H is generic and not everywhere non-constant, we proved above that we have an induced foliation of the compact submanifold N ⊂ E. If some super critical leaf L p were not compact, then L p would be contained within some open Novikov component V ⊂ N . Recall that a Novikov component V is a saturated open submanifold such that every leaf contained in V is non-compact and approaches some boundary leaf of its closure V . Moreover, from the proof of Lemma 3.11 all leaves in V , including the boundary ones in V should be totally geodesic. We are in position to apply now the following result of Carrière and Ghys ([9]) on totally geodesic foliations: Proof. -The codimension-one foliation of V is transversely orientable and totally geodesic. According to [9, Prop. II.2] such a foliation with a compact leaf is a fibration over an interval.
But this leads us to a contradiction, as we supposed V to be a Novikov component. This ends the proof of Proposition 3.3 in the case when H is generic and not everywhere non-constant. Remark 3.19. -The present proof does not extend to the case when H is eventually constant. There exist for instance totally geodesic foliations all whose leaves are noncompact and have constant curvature, as the ones described in [9] in dimension 3.
3.8. Uniform geometry of curvature leaves. -In this section we restrict ourselves to locally 1-QC manifolds. With notation from above we set: Then curvature leaves are totally umbilical with principal curvatures equal to α V . Note that if ξ| Lp is orientable then the curvature leaf L p is 2-sided and there exists a coherent choice of the lifts ξ V such that the value of α V (x) is independent on V , for any x ∈ L p . However, if ξ| Lp is non-orientable only the absolute value |α V (x)| is well-defined globally, as the curvature leaf is 1-sided. Moreover, from [6, (3.5)] and [22, (3 Recall that the focal radius of L at a point p ∈ L is the smallest r for which r ξ| p is a critical point of the exponential map and the focal radius f L⊂M is the smallest focal radius at p among all p ∈ L. Proof. -The first claim is the content of [20,Lem. 3]. The second claim is well-known (see [21,Lem. 2.3]). An explicit lower bound for the focal radius of L was provided by Warner in [53,Cor. 4.2], namely the focal radius of the totally umbilical hypersurface of principal curvature min α within the space of constant curvature max(H, N ).
For every p ∈ M C we consider the metric disk D γ (p) ⊂ L p of radius γ within the leaf L p endowed with the induced metric. Let i p,γ denote the normal injectivity radius of D γ (p) inside M , i.e., the largest r such that the restriction of the exponential map to the radius r disk subbundle of the normal bundle of L p (restricted to D γ (p)) provides a diffeomorphism onto its image.
We say that an arc ζ joining p, q ∈ L is an orthogeodesic if it is a minimal geodesic in M between p and q which is orthogonal to L at its endpoints. We denote by o Dγ (p)⊂M the smallest orthogeodesic length between points of D γ (p). It follows from [29,Th. 4.2] that the normal injectivity radius i p,γ is given by Our next goal is to show that i p,γ is bounded from below. We will closely follow the reasoning of Corlette from [16].
Let d L denote the Riemannian distance on the leaf L and d M the Riemannian metric on M . We denote by D M r (p) ⊂ M the radius r metric disk on M with center at p. It will be convenient for us to change the metric into a flat metric in a small neighborhood of a given point. The exponential map exp p : T p M → M restricts to a diffeomorphism from the tangent metric ball {v ∈ T p M ; v γ} onto D M γ (p). Let g denote the pull-back by exp −1 p of the flat Riemannian metric on T p M and d M the associated distance. According to [16,Lem. 2.1] and the lines after its proof p. 157, by choosing γ sufficiently small, we can arrange that the distorsion between g and the initial metric g be uniformly bounded: -For any λ > 1, δ > 0 there exists some R(λ, δ) (smaller than the injectivity radius) with the property that: (2) the difference between the two Levi-Civita connections on these two metric balls is bounded by δ (in the C 0 topology).
The second ingredient needed is the following version of a theorem of Schwarz, as stated by Corlette in every dimension: -Let c be a curve in the Euclidean space joining the points q and q and whose second fundamental form is bounded in norm by 1/r, where r is at least half the distance between q and q . Consider the circle of curvature 1/r which passes through q and q , which is separated by the pair of points into two arcs, say of lengths 1 2 . Then, either the length of c is smaller than 1 or else it is larger than 2 .
We are ready to prove now: Then there exists some γ depending only on A and the sectional curvature bounds such that the normal injectivity radius i p,γ of curvature leaves in E is uniformly bounded from below.
Consider an orthogeodesic β of length 2t joining the points q, q ∈ D γ (p). It is enough to consider that β is the shortest one, namely the midpoint u of β has d M (u, x) t, for every x ∈ D γ (p). Let c be a geodesic on L joining q and q , which is therefore contained in D γ (p) ⊂ D M γ (p). Observe that the second fundamental form of the geodesic c is bounded in norm by A.
Consider next the flat metric g on D M R(λ,δ) (u). Then the metric sphere ∂D M t (u) becomes an ordinary sphere of radius t in the Euclidean space. The second fundamental form of c in the flat metric g is bounded by 2A, by Lemma 3.21. The shortest arc of circle joining q to q of curvature equal to 2A has length 1 = arcsin(2At)/A.
Suppose that 100t < γ. Then 2At < 1/100 and so 1 < 2.2 t. On the other hand the curve c lies outside the sphere of radius t and hence its length is at least the length of a great arc of circle joining q and q on the sphere, namely πt.
From the Schwarz Theorem 3.22 it follows that the length of c must be at least the length 2 = (π/A) − 1 of the longest arc of circle joining q to q of curvature equal to 2A. Observe now that: Therefore the curve c in the flat metric should have length greater than 2.2 γ, contradicting the fact that d L (q, q ) 2γ and the Lipschitz constant is λ < 1.1.
We derive that t 1 100 γ and hence  (1) Let E be a connected component of M C with the property that λ| E is not identically zero. Then both the volume and the diameter of a curvature leaf L ⊂ E are uniformly bounded from above and from below away from zero.
(2) Let E be a connected component of M C with the property that λ| E ≡ 0. Then the volume of a curvature leaf L ⊂ E is uniformly bounded from below away from zero.
Proof. -By Proposition 3.1, E is diffeomorphic to an interval bundle over some closed space form P . If λ > 0 or λ < 0 and dimension n 4, then constant curvature metrics on P are unique up to isometry. In particular, the volume of L is a constant times λ −n/2 and the diameter is also a constant times λ −1/2 . As λ is bounded, the first claim follows. By Corollary 3.2 the only other possibility is that λ ≡ 0 on E. Moreover, if this is the case, then by Lemmas 3.20 the injectivity radius is bounded from below by γ and hence the volume of L is at least the volume of the Euclidean ball of radius γ. The next step is to construct uniform cylindrical neighborhoods around point of M . For this purpose, for every p ∈ M C we now consider the relative metric neighborhood N γ (D γ (p)), which is the image of the set {(q, ρξ) ; q ∈ D γ (p), |ρ| γ} ⊂ T M by the exponential map. Since the exponential map is a diffeomorphism on its image, the later is a relative regular neighborhood with respect to its boundary. Let now q ∈ M at distance γ/20 from D γ (p) the point which belongs to the orthogonal geodesic θ p to L p issued from p in the direction given by a lift ξ.
Lemma 3.26. -There exists some constants Q, R such that for γ < R the metric disk D M (q, Qγ) is contained within N γ (D γ (p)).
Proof. -We change the metric in D M (p, 4γ), with γ small enough, to the flat metric such that the second fundamental form is bounded by 2A, using Lemma 3.23. We can also assume that the line field ξ is defined on D M (p, 4γ) and that in the flat metric the angle between ξ| p and ξ| y is smaller than π/10, for any y ∈ D M (p, 4γ). Let p be a point of the metric sphere ∂D γ (p) ⊂ L p and θ be the orthogonal geodesic arc of length γ issued from p . By Lemma 3.23, N γ (D γ (p)) is diffeomorphic to a cylinder D γ (p) × [0, γ] whose lateral surface corresponding to ∂D γ (p) × [0, γ] is the union of all arcs θ p . Since the second fundamental form of D γ (p) is bounded, we can suppose from Lemma 3.24, by passing to a small γ if needed, that d M (p, p ) Bd(p, p ). In the flat metric the geodesics arcs θ p and θ p are line segments and their angle is smaller than π/10. Then elementary geometry arguments show that d M (q, θ p ) > 9 10 d M (p, θ p ), d M (p, θ p ) cos (π/10) d M (p, p ) and so d M (q, θ p ) > B 2 d(p, p ). This implies that the distance between q and the lateral surface of ∂N γ (D γ (p)) is at least B 2 γ. On the other hand a similar argument shows that the distance between q and the upper cap of the cylinder corresponding to D γ (p) × {γ} is also at least B 2 γ, which implies our claim for Q = B/2. (1) the source X is the union of closed manifolds, namely compact manifolds with empty boundary; (2) f has no triple points; (3) if Y is any (topological) connected component of the manifold X, then the restriction f | Y : Y → M is an embedding; (4) double points are tangencies between images of distinct connected components of X. Specifically, if (x 1 , x 2 ) ∈ X × X and f (x 1 ) = f (x 2 ), then there exist two distinct connected components X 1 , X 2 ⊂ X of X such that x 1 ∈ X 1 , x 2 ∈ X 2 and f (X 1 ) and The connected components of X are called the branches at the source, their images in M are the branches of f (X) and the set of double points in M is the branch set of f .
We extend this definition to smooth branched hypersurfaces with self-tangencies in M by allowing the restrictions f | Y : Y → M to be immersions with self-tangencies instead of embeddings.
By abuse of language we will also refer to the image f (X) as a branched hypersurface. With this convention the tangent space to a branched hypersurface is well-defined also at branch points.
Recall that any smooth hypersurface could be defined locally as the graph of some smooth function. This permits to define limits of hypersurfaces as follows. with the property that ψ i,n converges in the C k -topology to ψ i,∞ for all k 2.
Further, the sequence of submanifolds H n limits to the finite union We know that the distribution D is integrable on M C. Our aim is to show that a weaker version of integrability holds on the closure M C. Our main result in this section is: (1) Then for every q ∈ M C ∩ C there exists a limit leaf L q for D.
(2) If ξ is orientable then the limit leaf L q passing through q ∈ M C ∩ C is a branched hypersurface. Moreover, each branch of the limit leaf L q is a totally umbilical compact submanifold of constant intrinsic sectional curvature λ tangent to (the extension to M C of) the horizontal distribution D.
(3) When ξ is non-orientable, the limit leaf L q is a branched hypersurface as above, possibly with self-tangencies.
(4) Every limit leaf L q is contained within M C ∩ C.
Outline of the proof of Proposition 3.29. -We have to analyze the behaviour of leaves L pi , when p i approach a point q ∈ C ∩ M C. We first consider the problem locally. Using the bounded geometry of leaves from Section 3.8 we shall consider specific neighborhoods which are products of a disk in a leaf and an interval. The size of these neighborhoods is bounded from below and chosen small enough such that leaves intersections are graphs of functions. The Arzelà-Ascoli argument provides the existence of a lower and an upper limit for the corresponding sequence of functions, whose associated graphs form the two local branches passing through q. An uniform lower bound for the normal injectivity radius of leaves permits to show that each leaf L pi intersects only once (or twice in the non-orientable case) a specific neighborhood. This permits to pass from a local limit leaf to a global limit leaf which should be compact since it can be approached everywhere by some sequence of compact leaves L pi . Proof.
We can take A large enough such that max(|H(x)|, |N (x)|) A, for all x ∈ M . Since every curvature leaf L ⊂ M C is totally umbilical of principal curvatures |α|, the previous inequality shows that the second fundamental form II L has uniformly bounded norm: By hypothesis there exists a local conformal change of the metric to a flat metric, in particular we can assume that there is a flat metric on each V j . Changing the original metric on V j to the flat one amounts to changing the constant A above. Now, we can therefore compare tangent vectors at different points of V j . As ξ| V j is a smooth vector field defined on V j , there exists δ(ε) > 0 such that whenever x, y ∈ V j are at distance at most δ apart then the angle between ξ| x and ξ| y is at most ε.
Our key ingredient is the following lemma which seems to be well-known to the specialists. (1) The size of U q is uniformly bounded from below, namely it contains an interval bundle over a metric disk having both radius and height larger than some constant independent on q.
(2) The intersection of the curvature leaf L pi with U q is connected if ξ is orientable and has at most two connected components, if the restriction of ξ is to the connected component of M C containing p i is non-orientable.
(3) Let (L pi ∩U q ) • denote the connected component of L pi ∩U q containing p i . Then the sequence (L pi ∩ U q ) • limits to a branched hypersurface (L q ∩ U q ) • , called a piece of the local limit leaf at q. Moreover, (L q ∩ U q ) • is the union of two codimension 1 submanifolds of M passing through q which are tangent to D which are called (local) branches.
(4) For any sequence of points p i ∈ M C with lim p i = q, the sequence (L pi ∩U q ) • limits to (L q ∩ U q ) • or a branch of it.
(5) Each piece (L q ∩ U q ) • of the local limit leaf at q is contained in (M C) ∩ C.
(6) The sequence of curvature leaves L pi ∩ U q limits to a local limit leaf L q ∩ U q . Moreover, the local limit leaf L q ∩ U q is the union of at most two pieces of local limit leaves.
Proof of Lemma 3.30. -According to Lemmas 3.20 and 3.23 there is γ such that any leaf L has both injectivity radius i L and focal radius f L⊂M bounded from below by γ while the normal injectivity radius i p,γ is bounded from below. Now, as II L A and each leaf is of constant curvature α, uniformly bounded, it follows that each the embedding of each leaf L → M has uniformly bounded local metric distorsion. Change now γ into min(γ, i p,γ ). It follows that i p,γ γ.
Consider now the image N γ (D γ (p)) of {(q, ρξ) ; q ∈ D γ (p), |ρ| < γ} ⊂ T M by the exponential map. By above the exponential map is a diffeomorphism on its image, which is a relative regular neighborhood with respect to its boundary. Moreover, as We can assume that Note that there is a projection on the first factor π : U q → D γ (p 1 ) ⊂ L p1 . Specifically, π(z) ∈ D γ (p 1 ) is the closest point to z. By our choice of γ, the map π is well-defined (see Lemma 3.23). Moreover, we can suppose that q, p i ∈ int(U q ), for all i and that U q ⊂ V j .
We will suppose henceforth that ξ is defined on a γ-metric neighborhood Ω of M C, and in particular on every U q , for q ∈ M C ∩ C.
By taking a smaller γ if needed, we can assume that U q is included in a metric ball of radius δ(π/10), namely that ξ| x has only a small angle variation for all x ∈ U q and moreover d(p i , p 1 ) < γ/10. Now, U q satisfies the requirements of the first claim in the lemma, by construction.
From Proposition 3.1 every connected component E of M C is an interval bundle over a compact space form P , as E is a cylinder. Therefore the complement of the lift of every curvature leaf in E has two connected components. Then [28,Lem. 2,p. 385] shows that under these conditions any transversal arc will cut a leaf of E in at most one point. On the other hand, note that a geodesic arc issued from a point of D γ (p 1 ) and orthogonal to L p1 should be transversal to all leaves that it intersects in U q , because of the small angle variation assumption. If ξ is orientable, then the intersection of curvature leaves L pi with U q to should be connected, as otherwise orthogonal geodesics will intersect twice the same leaf. If ξ is non-orientable, the same argument works for the pull-back of U q to E should be connected and hence the intersection of curvature leaves L pi with U q have at most two connected components. This proves the second item of our Lemma.
We further claim that the connected component of (L pi ∩ U q ) • is the graph of a smooth function y i : D γ (p 1 ) → [−γ, γ]. This is clear for (L p1 ∩ U q ) • and hence for all leaves (L pi ∩ U q ) • which are close enough to it. We have to prove that this holds for all points p i with d(p i , p 1 ) < γ/10.
A compact hypersurface L ∩ U q is the graph of some smooth function D γ (p 1 ) → [−γ, γ] if and only if the projection π : U q → D γ (p 1 ) restricts to a diffeomorphism π| L : L ∩ U q → D γ (p 1 ). The key point is to show that π| L is a bijection, as the smoothness of its inverse will follow from the smoothness of L ∩ U q .
If π| Lp i were not surjective, then there are two possibilities: -either (L pi ∩ U q ) • would intersect the top or the base of the cylinder U q , namely the image of D γ (p 1 ) × {−γ, γ}; -or else a boundary point of the image π| Lp i ((L pi ∩ U q ) • ) would be a critical value of π| Lp i lying in int(D γ (p 1 )).
In the first case we can find a curve contained in (L pi ∩ U q ) • joining p i to some point z ∈ D γ (p 1 ) × {−γ, γ}. By the mean value theorem there is some point r on this curve where the angle between ξ| p1 and ξ| r is at least arcsin ( √ 199 − 1)/20 > π/10, contradicting our hypothesis on the small variation of the angle in the (flat) chart U q .
In the second case (L pi ∩ U q ) • only intersects non-trivially the lateral surface ∂U q D γ (p 1 ) × {−γ, γ} of the boundary ∂U q . Now, if x is a critical point of π| Lp i then ξ| x should be parallel to the tangent space T π(x) D γ (p 1 ) and hence orthogonal to ξ| π(x) , contradicting our assumptions on the small variation of the angle. We conclude that π| Lp i is surjective.
Next, if π| Lp i were not injective when restricted to a connected component of (L pi ∩ U q ) • , then there would exist a smooth path in (L pi ∩ U ) • joining two points with the same image under π| Lp i . The restriction of π| Lp i to this path must then have a critical point and, as above, this would provide a point x of (L pi ∩ U q ) • for which ξ| x is orthogonal to ξ| π(x) . This would contradict our assumption on the small variation of the angle. Our claim follows.
Observe that y i : D γ (p 1 ) → R are smooth bounded functions, as |y i | γ. Their derivatives are also bounded as ξ| x is close to ξ| q , for any x ∈ U q , by the small angle variation assumption. Specifically, assume that we are in a flat chart containing U q obtained by a conformal change of metric. We pick up coordinates {x 1 , x 2 , . . . , x n−1 , y}, the first n − 1 coordinates corresponding to D γ (p 1 ). If the point x ∈ U q belongs to the image of y i , then we have: from which we derive the following upper bound for the first derivatives of y i : n−1 j=1 |∂y i /∂x j | 2 < sin 2 (π/10) 1 + sin 2 (π/10) < 0.1.
In the same flat chart the second fundamental form of the hypersurface given by the image of y i has the matrix: Since the norm of the second fundamental form of L pi ∩ U q is uniformly bounded by A, we derive the following bound for the second derivatives of y i : Alternatively, recall that II is diagonal in the 1-QC metric, because leaves are totally umbilical and the principal curvatures equal α. The entries of II in the flat chart can be expressed in terms of the entries of II in the 1-QC metric (namely 0 and α) and the real function specifying the conformal change in the metric. Since all derivatives of α are bounded in U q higher derivatives of y i are also bounded. By the Arzelà-Ascoli theorem, the sequence y i has a sub-sequence which converges to a smooth function y ∞ : D γ (p 1 ) → [−γ, γ] in the C k topology, for every k 2. Now, the leaves L pi are pairwise disjoint and hence there is a total order among the functions y i . Therefore, there are only two possible limits of sub-sequences of y i , namely y + ∞ = lim sup y i and y − ∞ = lim inf y i . We say that the sequence (L pi ∩ U q ) • is increasing or decreasing if the sequence of functions y i is increasing or decreasing, respectively.
Denote by (L ± q ∩ U q ) • the graph of the limit function y ± ∞ and call them the local branches of the local limit leaf. We say that the sequence L • pi ∩ U q is below q if the sequence y i is increasing and above q if the sequence y i is decreasing. Each local branch is a smooth totally umbilical hypersurface of principal curvature α(q). Since ξ is everywhere orthogonal at L pi , by passing to the limit we find that each (L ± q ∩ U q ) • is a codimension 1 submanifold which is tangent to the distribution D at each point • for the union of the two local branches. The local branch set is the set of tangency points between the two local branches (L ± q ∩ U q ) • . The branch set could be all of (L q ∩ U q ) • , in which case the two local branches coincide, or just one point.
Eventually, observe that (L pi ∩ U q ) • limits to (L q ∩ U q ) • since the later is the union of the graphs of the functions lim sup y i and lim inf y i .
We obtained above a piece (L q ∩U q ) • of the limit leaf at q. We will need now to show that the piece (L q ∩ U q ) • does not depend on the choice of the sequence p i ∈ M C with lim p i = q. It is enough to consider the case when the sequences p i and p i are both increasing with respect to the order defined in the proof of Lemma 3.30. As their limit is the same the sequences of functions y i and y i associated to the sequences p i and p i , respectively, are comparable, namely for every i 1 there exists some i 2 , j 1 , j 2 such that y i1 y j1 y i2 y j2 . Then lim y i = lim y i , which implies our claim and hence item (4) of our Lemma.
Further, if q ∈ M C ∩ (L q ∩ U q ) • , then (L q ∩ U q ) • = (L q ∩ U q ) • , so that q ∈ L q . But the curvature leaf L q is complete with its intrinsic metric, according to Reckziegel's Theorem 3.4 (see Lemma 3.5) and L q ⊂ M C. This leads to a contradiction. Thus (L q ∩ U q ) • ⊂ C. On the other hand every point of (L q ∩ U q ) • is a limit point of (L pi ∩ U q ) • , so that (L q ∩ U q ) • ⊂ M C.
We can define now the local limit leaf L q ∩ U q , in the case when L pi ∩ U q is not connected, for large enough i. It suffices to consider the set p i ∈ M C ∩ U q with the property that L pi ∩ U q = (L pi ∩ U q ) • (L p i ∩ U q ) • and lim p i = q ∈ C. Then we set L q ∩ U q = (L q ∩ U q ) • ∪ (L q ∩ U q ) • . It follows from above that L pi ∩ U q limits to L q ∩ U q . Such a local limit leaf L q ∩ U q is obviously unique and independent on the choice of the sequence p i . Moreover, by uniqueness if q ∈ C is such that q is a limit Proof. -Consider a normal orthogeodesic joining p, q ∈ L p . We consider the metric neighborhood N γ (D γ (p)) of p. Then the argument used above to prove Lemma 3.30 (2) shows that L p ∩ N γ (D γ (p)) is connected. In particular d M (p, q) γ. This provides an uniform lower bounds for the length of an orthogeodesic. Using [29,Th. 4.2] and the lower bound for the focal radius provided by Lemma 3.20 we obtain the claim.
We now define an equivalence relation, called confluence, for branches of local limit leaves as follows. First, if U q ∩ U q = ∅ we say that L + q ∩ U q and L + q ∩ U q have a common extension if (L + q ∩ U q ) ∩ U q = (L + q ∩ U q ) ∩ U q . Then L + q ∩ U q and L + q ∩ U q are called confluent if there exists a finite sequence of branches of local limit leaves starting at L + q ∩U q and ending with L + q ∩U q , where consecutive terms have a common extension.
We now define the branch L + q of the limit leaf passing through q as the union of all branches of local limit leaves confluent with L + q ∩ U q . By definition L + q is an immersed smooth codimension-one submanifold of M with possible self-tangencies, which is contained within M C ∩ C. Eventually, we define the limit leaf L q as the union of all branches of all local limit leaves confluent to some branch of L q ∩ U q .
Lemma 3.32. -Each branch L + q of a limit leaf is compact.
Proof. -We want to show that limit leaves can be approached globally by leaves.
The following property of the confluence relation will be the key ingredient of the proof. We suppose that: (1) L pi ∩U q limits to L + q ∩U q , where p i are below q, and L p i ∩U q limits to L + q ∩U q ; (2) U q ∩ U q = ∅; (3) L + q ∩ U q and and L + q ∩ U q have a common extension. Then, we claim that there is a sub-sequence L p i of the sequence of leaves L pi , L p i with the property that L p i ∩ (U q ∪ U q ) limits to (L + q ∩ U q ) ∪ (L + q ∩ U q ). Observe that p i are below q and p i are below q . Our claim is a consequence of an analogous result on sequences of real functions y i , having the property that y i+1 − y i have constant sign. Specifically, assume we have two increasing sequences of real functions y i , y i : U ∪ V → R such that their restrictions to the open relatively compact sets U and V , with U ∩ V = ∅, converge to the restrictions of a function z : U ∪ V → R defined on U ∪ V , namely y i | U → z| U and y i | V → z| V . Moreover, suppose that y i −y j have constant sign, for all i, j. Then there exists a sub-sequence y i of the sequence obtained by putting together y i , y i such that y i | U ∪V → z| U ∪V . In fact it suffices to take y i = max(y i , y i ), which coincides with either y i or y i on U ∪ V . To allow a description of general hypersurfaces which might not be globally graphs we should extend this result to multi-valued functions. Consider now that y i | U , y i | V , z| U and z| V are now usual functions, but y i | V and y i | U are only required to be multi-valued functions whose graphs correspond to some smooth hypersurface. The argument above then shows that the graphs of the possibly multi-valued functions max(y i , y i )| U ∪V converge to the graph of z| U ∪V in the C 0 -topology: if y i | U ∩V > y i | U ∩V , then max(y i , y i ) = y i and y i | V < y i | V < z V . Thus the limit of any subsequence y i | V as above is z| V . However, if we assume that all derivatives of y i | V and y i | V are continuous then y i | V should be also usual functions, as otherwise their first derivative would be infinite in some direction. In our case of branches of limit leaves this assumption is satisfied since all leaves intersecting a open distinguished neighborhood U q have almost horizontal tangent space.
By induction on the number of local branches we obtain that, up to extracting a sub-sequence (and possibly changing the initial sequence p i by another sequence with the same properties) L pi limits from below to L + q . Each branch L + q is complete with respect to the induced metric. To see that, it suffices to note that the metric disk D Qγ (q) within the branch L + q is contained within L + q ∩ U q , for every q, according to Lemma 3.26. To prove that L + q is compact, it is therefore enough to show that it is a proper leaf.
We assume that in our construction above the height γ of each U q is smaller than i p,γ /3. If U q = N γ (D γ (p 1 )) we denote by U q the corresponding metric neighborhood of double height, namely N 2γ (D γ (p 1 ). Since L + q is covered by some union U q , its closure L + q in M is covered by the corresponding union U q . By compactness we can extract a finite covering U qi of L + q . If L + q were non-compact, then there would exist some neighborhood U q = N 2γ (D γ (p 1 )) which intersects infinitely many times L + q . But U q has the shape of a cylinder, so that there are geodesics issued from D γ (p 1 ) ⊂ L p 1 orthogonal to L p 1 which intersect infinitely many times L + q . Recall from above that every point of L + q can be approached arbitrarily closely by the leaves L pi . It follows that there exist leaves L pi which intersect more than 100 times some orthogonal geodesic within U q . This contradicts the fact that any transverse arc in M C should intersect at most twice every leaf of our foliation within a small neighborhood (see [28,Lem. 2,p. 385] and Lemma 3.30 (2) above).
From above L q is the image of a codimension 1 immersion of some union X of closed manifolds, with only tangency or self-tangency points. The norm of the second fundamental form of every branch is also bounded by A, as this bound is valid for any local branch L ± q ∩ U q .
When ξ is orientable, then every branch is embedded, as the normal injectivity radius is bounded from below by Lemma 3.31 and hence there are not self-tangency points. Thus L q is a branched hypersurfaces. Further L q ⊂ M C∩C and each branch is a totally umbilical hypersurface with constant sectional curvature λ(q). When ξ is non-orientable, then the lift L q to the double cover Ω is a branched hypersurface, as claimed. This proves Proposition 3.29.
An immediate corollary of the proof is the following: Then, up to passing to a sub-sequence, the limit of L pi in the Hausdorff topology is some branch of the limit leaf L q . More specifically, if p i are below q then the limit of L pi is the branch L + q .
Recall that the Hausdorff distance between the non-empty compact subsets X, Y ⊂ M is given by max(sup x∈X inf y∈Y d(x, y), sup y∈Y inf x∈X d(x, y)), where d is the distance in M . It is well-known that the space of compact subsets of M equipped with the Hausdorff metric is complete.

Closure of non-compact globally 1-QC components
We denote by Fr(E) = Fr(E) = E E the frontier of a subset E ⊂ M . An end of a topological space X = ∞ n=1 K n which is the ascending union of compact subspaces K n , is an infinite sequence V 1 ⊃ V 2 ⊃ V 3 ⊃ · · · with n V n = ∅, where each V n is a connected component of X K n . When X is a path-connected CW-complex ends correspond to equivalence classes of proper continuous maps R + → M , up to proper homotopy of their restrictions to the subset of positive integers. For example R has two ends, R 2 has one end while the infinite complete binary tree has uncountably many ends. Proposition 3.29 shows that limit leaves are branched surfaces hence union of embedded (or immersed) compact hypersurfaces. Here we will obtain a more precise statement by showing that the number of branches in the frontier of a connected component E of M C is either one or two. Proof. -By Proposition 3.1 E is diffeomorphic to a cylinder P × R over some closed space form P . It makes sense then to say that a sequence of curvature leaves which goes to infinity belongs to one or another of the ends of R.
Let q ∈ Fr(E). By Proposition 3.29 there exists a limit leaf L q passing through q, which is a compact branched hypersurface everywhere tangent to D. Now L q ∩ E ⊂ Fr(E) as L q ∩ (M C) = ∅. Note that an arbitrarily small backward trajectory of ξ issued from q cannot be contained in L q , as ξ is orthogonal to L q . Then, according to Lemma 3.33, any sequence of curvature leaves belonging to a given end of E converges to a branch of L q ∩ E. Thus L q has at most two branches, corresponding to the two ends of E. Therefore Fr(E) is a compact branched hypersurface with at most two branches.
If there is only one branch for each end, then E is a compact manifold endowed with a foliation with compact leaves and hence diffeomorphic to a cylinder. Otherwise, E is homeomorphic to the result of gluing together the two ends of the cylinder along the branch locus, namely the quotient of P × [0, 1] by the equivalence relation (x, 0) ∼ (x, 1), for every branch point x. Proof. -Let p E : E → E denote the projection. We can write ϕ(x, 0) = (φ(x), 1), for any x ∈ P , where φ : P → P is a diffeomorphism. As ξ is locally orientable, we can consider the trajectory γ of ξ in E issued from the point p E ((x, 0)) = p E ((φ(x), 1)). Then p −1 E (γ) is the union of two arcs, say γ (x,0) and γ (φ(x),1) which are respectively issued from (x, 0) and (φ(x), 1). However, the pull-backs of ξ-trajectories are now trajectories of ξ, in particular they are horizontal, i.  ((φ(x), 0)). If we travel backward along γ starting from p E ((φ(x), 0)) we have to end at p E ((φ, 0)); we derive that ϕ(φ(x), 0)) = (x, 1), and thus φ(φ(x)) = x. Moreover, as ξ is a unit vector field we have more generally that p E ((x, t)) = p E ((φ(x), 1 − t)).
This shows that φ : P → P defines a fixed point free involution.
Set P = P /G, which is a closed manifold by Lemma 3.36 and let p P : P → P be the quotient map.
The projection map P × [0, 1] → P sending (x, t) into p P (x) factors through a map P × [0, 1]/ (x,t)∼(ϕ(x),1−t)) → P . The latter induces a well-defined map E → P , where P denotes the image of P by the immersion in M . The preimage of a point p P (x) ∈ P by the induced map E → P is the maximal trajectory arc γ x ⊂ E joining the points x and φ(x) from Fr(E). It is immediate that the map E → P is a fibration with fiber [0, 1], which is covered by an interval fiber bundle P × [0, 1]/ (x,t)∼(ϕ(x),1−t)) → P . The monodromy action of j sends the fiber γ x into itself by exchanging its endpoints, while elements from π 1 ( P ) act as identity. It follows that E → P is the twisted product P × α [0, 1], image by an immersion (and its equivariant lift) of the twisted product P × α [0, 1], where α is a class in H 1 (P ; Z/2Z).
On the other hand E → E is a 2-fold cover of class w 1 (ξ) ∈ H 1 (E; Z/2Z), by hypothesis. This cover is the image of an immersion of the 2-fold cover P × [0, 1] → P × α [0, 1]. Therefore its class in H 1 (P × α [0, 1]) is determined by the exact sequence: and thus it can be identified with α, so that α is the pull-back of w 1 (ξ| E ) to H 1 (P ; Z/2Z). Recall that E is diffeomorphic an open interval bundle over P . Thus this class can be identified geometrically with w 1 (ξ| E ) ∈ H 1 (P ; Z/2Z). This settles Proposition 3.35.
3.11. Limit leaves subspace. -The limit leaves subspace of a 1-QC manifold is the compact set L = Fr(M C) = M C ∩ C.
Components E of M C which are not cylinders will be called topologically nontrivial. Proof. -Let E be a topologically non-trivial component of M C. If E is a cylinder with its ends glued along a non-empty subset of the boundary, then the class of a loop which follows the vertical line contributes a non-trivial factor Z to H 1 (M ). If E is a non-trivially twisted cylinder then it contributes a non-trivial factor Z/2Z to H 1 (M ). Since M is compact H 1 (M ) is of finite type and hence we cannot have but finitely many such components.

Consider a maximal set B of cylindrical components
Then each E i contributes a non-trivial factor Z in H 1 (M ) which is of finite type and thus B is finite. If E is another cylindrical component E = P × R, then cutting M i E i along P will disconnect it. In particular the class of P is a sum of several classes among the P i in H n−1 (M ), or by duality in H 1 (M ). Thus, the set of homology classes of cores P of non-essential cylindrical components is finite. We pick up a set B of such representatives. It follows that for any cylindrical component E = P × R in M , either P bounds or there exists some Q ∈ B which is homologous to P and hence P ∪ Q bounds a codimension zero submanifold Z. Proof. -In the proof of Proposition 3.29 we noticed that there exists some γ > 0 such that the γ-neighborhood N γ (L) is diffeomorphic to a twisted product for any leaf L.
Consider now a leaf L bounding a submanifold Z and a point p ∈ Z at distance γ/2 from L. Such a point exists as N γ (L) is embedded in M . Then the metric ball of radius γ/2 with center p is diffeomorphic to a ball and contained in Z. By the Rauch comparison theorem there exists some constant v depending on M such that this metric ball has volume at least v and hence the volume of Z is bounded below by v.
Thus there are only finitely many disjoint submanifolds Z with boundary a leaf.
We add to the set B a maximal set of leaves which bound disjoint submanifolds.
Proposition 3.40. -There exists ε > 0 such that for any leaves Q ∈ B and P cobounding with Q a codimension zero submanifold of M , such that min p∈P,q∈Q d(p, q) < ε, then Z is diffeomorphic to a cylinder.
Proof. -We prove first a slightly weaker assertion, as follows.
Lemma 3.41. -Let Q, P be the cores of two leaves belonging to cylindrical components of M C. There exists δ > 0 such that for any leaves Q ∈ B and P cobounding with Q a codimension zero submanifold of M , such that P ⊂ N δ (Q), then Z is a cylinder.
Proof. -We use the finite cover {U i } by cylinders of the form int(U ) from the proof of Proposition 3.29 above. Then P ∩ U i and Q ∩ U i are both disjoint graphs of functions defined on U i , for every i. Therefore they are the upper and bottom cap of a cylinder on U i . The union of all these cylinders is a cylinder over Q, since ξ| Q must be orientable.
It remains to prove that for every δ > 0 there exists ε > 0 such that for any leaves Q ∈ B and P cobounding with Q a codimension zero submanifold of M , such that min p∈P,q∈Q d(p, q) < ε, then P ⊂ N δ (Q). Assume the contrary. Then we have a sequence of leaves Q i cobounding with P such that lim i min p∈P,q∈Qi d(p, q) = 0, but P ⊂ N δ (Q i ). This provides a sub-sequence Q i such that Q i converges to P but then P ⊂ N δ (Q i ) for all large enough i, contradicting Proposition 3.29 and Lemma 3.33. Proof. -By Proposition 3.29, L is the union of all limit leaves L q . Note that two limit leaves L q , L p ∈ L are either disjoint or coincide. In fact they cannot intersect transversely, as both are limits of sequences of foliations leaves and if they are tangent somewhere then they should coincide.
In the proof of Proposition 3.29 we noticed that there exists some γ > 0 such that the γ-neighborhood N γ (L p ) is diffeomorphic to a twisted product for any leaf L p . This continues to hold for any branch L * q of some limit leaf L q . We consider a maximal set F of leaves which are cores of cylindrical components E such that no two of them belong to the same component and min p∈P,q∈Q d(p, q) > ε/2, for every two leaves P and Q. By above there are only finitely many such leaves. Consider the collection of branched hypersurfaces associated to components E which are either topologically non-trivial or else they have leaves from F and denote by L their union. We claim that L verifies the claim. In fact the closures of the connected components of M L are either twisted cylinders or closures of components of int(C). In fact two cylinder components are separated by a component of int(C) or by a limit leaf. In the former case we need that the distance between boundaries be at least ε or otherwise the union of the three components will be a cylinder. In particular there are only finitely many such components of int(C).
Remark 3.43. -Note that L might have infinitely many connected components, for instance an infinite union of compact leaves accumulating on a compact leaf. Nevertheless these leaves are contained in int(M C).

3.12.
End of proof of Theorem 1.6. -According to Proposition 3.42, M is the union of some interval bundles and finitely many pieces X i where X i are connected components of int(C). Each X i is an open manifold with a constant curvature metric. Moreover, the curvature of each point of the closure X i ⊂ M is also constant. The frontier Fr(X i ) consists in finitely many components, each one being either a totally umbilical constant curvature submanifold, the union of two such submanifolds of M which are glued together along some subset in such a way that the tangent space is well-defined at any point of Fr(X i ) or else a totally umbilical constant curvature immersed hypersurface with self-tangencies.
Suppose that there exists a branched hypersurface L + ∪ L − in the frontier Fr(X) of some X as above. Since L + ∪ L − is a limit leaf, there exist arbitrarily small onesided neighborhoods of every L ∈ {L + , L − } which are diffeomorphic to cylinders, the other boundary components being curvature leaves. Thus, by adjoining finitely many cylinders to X we obtain a manifold with boundary X ext ⊃ X. We claim that X ext also admits a constant curvature metric with totally umbilical boundary components.
We first claim that there exists a smooth f : [0, 1] → R + such that the warped product metric on the cylinder Q = L × [0, 1] with rescaling function f has constant curvature H. Recall that L has constant intrinsic curvature H + α 2 and is totally umbilical. The curvature k Q (σ) of a tangent plane σ spanned by the vectors x + v and w, where v and w are tangent to L, x to the [0, 1] factor and x 2 + v 2 = 1 (in the warped product metric) is given by: where k L denotes the sectional curvature of L. It follows that for every values of H and H + α 2 there exists some scaling function f defined on a small interval containing 0 such that f (0) = 1 and k Q (σ) = H, for any σ (see e.g. [4,Cor. 7.10] when H +α 2 0). Now we can glue together the Riemannian manifold obtained by cutting M open along L + and gluing the cylinder Q + with its warped product metric of constant curvature H along L + . The result is a smooth Riemannian metric because the corresponding second fundamental forms of L + within M and Q + agree. We do a similar construction on Q − and further for every branched leaf in Fr(X).
Eventually, note that the warped metric formula also makes sense for the case where L + is an immersed hypersurface in M . As we only have self-tangencies along L + we can perform the gluing procedure as above.
The result is a Riemannian manifold diffeomorphic to X ext with a metric of constant curvature and totally umbilical boundary components.
Therefore M is a graph of space forms. This proves Theorem 1.6.

4.1.
Outline. -The positivity of λ implies that connected components of the set of non-isotropic points consist of spherical cylinders. In the second step we describe the closures of these connected components, which are not necessarily manifolds.
To simplify the topology we want to excise the cores of these tubes, although their number might be infinite. A key argument is Grushko's theorem in group theory which translates here into the finiteness of the number of factors in a free amalgamated product presentation of a finitely presented group. This shows that one could carry out surgery for only finitely many tubes. It remains to understand the topology of the remaining pieces, which are neighborhoods of the set of isotropic points. We show there is a way to cap off the boundary components of each piece, by means of hypersphere caps in the hyperbolic space whose geometry is controlled. A deep result of Nikolaev (see [44] and Proposition 4.20 for a precise statement) states that a manifold with bounded volume and diameter whose integral anisotropy is small enough should be diffeomorphic to a space form. This procedure constructs closed manifolds out of regular neighborhoods of isotropic points, whose geometry is controlled and whose integral anisotropy is arbitrarily small. Then Nikolaev's result shows that these are diffeomorphic to space forms.

4.2.
The topology of M C when λ > 0. -Let M n be a closed n-manifold, n 3, admitting a 1-QC metric with orientable distinguished line field, which is assumed conformally flat when n = 3. Suppose that λ is positive and π 1 (M ) is infinite torsionfree.
By Proposition 2.14 there exists an isometric immersion Φ : M → H n+1 κ . Note that each curvature leaf is mapped isometrically onto some round (n − 1)-sphere by the immersion in H n+1 κ (see e.g. [47]). Recall that round n-spheres (sometimes also called geodesic or extrinsic n-spheres) are intersections of a hypersphere with H n κ ⊂ H n+1 κ (see [47, §2]). Now, any curvature leaf L in M is covered by a disjoint union of spheres in M and thus L is diffeomorphic to a spherical space forms. The stabilizer of one sphere in M is finite and obviously contained in π 1 (M ). Thus it must be trivial as π 1 (M ) was assumed to be torsion-free and hence the curvature leaf L is a sphere. Therefore, E is diffeomorphic to S n−1 × R.
When C = ∅ then, M is diffeomorphic to a spherical bundle over the circle.

4.3.
Closures of globally 1-QC components within 1-QC manifolds. -Suppose now that there exist isotropic points, so that C = ∅.
We assume that λ c > 0 on E. Then E is obtained from E by adjoining two boundary spheres, which might possibly be degenerate or tangent at one point or along a codimension-one sphere.
Proof. -The proof is similar to [8,Lem. 2.7]. Let q ∈ E ∩C. Choose a component E 0 of π −1 (E) ⊂ M , which is diffeomorphic to a spherical cylinder since E is simply connected. Let then q be a lift of q which belongs to E.
Any connected component L p of the preimage π −1 (L p ) ⊂ M of a curvature leaf through p is sent by the isometric immersion Φ onto an embedded round sphere S λ(p) of curvature λ(p) in H n+1 κ . Now, L p is a quotient of the sphere L p by a finite subgroup of π 1 (M ). Since π 1 (M ) is torsion-free L p is also a sphere, which is isometric to S λ(p) .
Let p i ∈ E be a sequence converging in M to q and p i ∈ E 0 lifts converging to q in M . The curvature leaves L pi lift to curvature leaves L pi in M which are round spheres of radius R i . Since λ i is bounded from below, R i are bounded. The sphere L pi lives in a geodesic n-plane H n κ orthogonal at the lift of the unit vector field ξ| pi . By a compactness argument and passing to a sub-sequence we can assume that R i and ξ| pi converge to R ∞ (which might be 0) and ξ ∞ , respectively. Then the spheres L pi converge to the round sphere L ∞ of radius R ∞ living in the n-plane H n κ orthogonal to ξ ∞ and passing through q.
The argument from [8,Lem. 2.7] applies to show that for any sequence p i ∈ E 0 converging to q in M the spheres L p i converge to the round sphere L ∞ . Indeed, if we had another limit sphere L ∞ passing through q, then a small enough connected neighborhood V of some point p ∈ L ∞ L ∞ would intersect all leaves L pi and would miss all L p i . Since any curvature leaf in E 0 disconnects E 0 , this would imply that V is not connected, contradicting our choice.
Therefore E 0 is obtained from E 0 by adjoining two boundary (possibly degenerate) round spheres corresponding to the ends. Since E 0 is compact and π is a covering π(E 0 ) = E. Then E is obtained from E by adjoining two round spheres. They cannot intersect transversely, as otherwise close enough leaves of E should intersect. Thus the boundary spheres should be either disjoint or tangent. Lifting both of them to M we obtain two round (n − 1)-spheres in H n+1 κ . They bound two convex n-balls whose intersection is convex. Therefore the spheres intersection is either one point or a (n − 2)-sphere and in particular it is connected.
If one only assumes that the metric is 1-QC the curvature leaves function λ might have poles at C, contrasting with the situation described in Lemma 2.11 for locally 1-QC metrics. The behavior of λ is directly related with the behavior, possibly wild, of the distinguished line field ξ in a neighborhood of an isotropic point. Consider a spherical cylinder with a ball adjoined to it, having the form of a rotationally invariant cap whose north pole is the only isotropic point. Then curvature leaves are spheres whose intrinsic curvature λ explodes when approaching the isotropic point. In fact, this is the only phenomenon which can arise:

Then,
(1) either there exists a connected E component of M C with the property that q ∈ E and E is diffeomorphic to a ball containing q, (2) or else there exists a sequence E i of components of M C with the property that q ∈ ∞ i=1 E i , but q ∈ E i , for any i. Moreover, E i are embedded annuli with disjoint interiors.
Proof. -Given a compact Riemannian M there exists some constant c 1 such that any embedded sphere S of diameter d c 1 bounds an embedded ball D of diameter f 1 (d) and volume f 2 (d), such that lim d→0 f i (d) = 0, i = 1, 2. It suffices to cover M by finitely many open charts diffeomorphic to balls and to choose then c 1 so that any metric ball on M of radius c 1 is contained within some chart. Now, the diameter of a round sphere S λ goes to 0, when λ → ∞. The same holds for the diameter of L λ in the Riemannian metric of M . Therefore, if λ is larger than some c 2 then L λ bounds an embedded ball D λ in M .
Consider λ 1 such that the intrinsic diameter of S λ1 is smaller than c 1 . There exists a curvature leaf L 1 of curvature 2λ 1 in E, which therefore bounds a ball D 1 ⊂ M . Let L 2 ⊂ E ∩ (M D 1 ) be a curvature leaf whose curvature is λ > λ 1 and thus it bounds also a ball D 2 in M . The leaves L 1 and L 2 bound a submanifold E 0 . From Proposition 4.1 and the strong form of the global Reeb stability theorem (see [43,18]) E 0 is compact and diffeomorphic to a spherical cylinder.
Observe that there is an unique embedded ball D bounded by an embedded codimension-one sphere in M , as otherwise M would be homeomorphic to a sphere, contradicting the fact that π 1 (M ) is infinite. Hence D 2 = E 0 ∪ D 1 .
If q ∈ D 1 , as q ∈ C ∞ we could choose λ large enough so that f 2 (λ) < vol(D 1 ). On the other hand, D 2 ⊂ D 1 and hence the volume of the unique embedded ball bounding L 2 would be larger than vol(D 1 ), contradicting our choice of λ. Thus q ∈ D 1 .
Note that, when q ∈ C ∞ is not as above, then there exists an infinite sequence E i of connected components of M C such that q ∈ i E i , but q ∈ E i , for all i. These components should have disjoint interiors and they are diffeomorphic to spherical cylinders. The size of E i (namely the radius of a boundary circle and the width) should converge to 0. Moreover, for every ε > 0 all but finitely many E i are contained in a metric ball of radius ε around q. They are therefore null-homotopic. Then there exist arbitrarily small saturated neighborhoods of C, namely for every ε > 0 small enough there exists an open neighborhood W ε ⊂ V ε (C) such that W ε C is a union of leaves and W ε ⊃ V δ (C).

Construction of neighborhoods of isotropic points
Proof. -This is a classical result in codimension-one foliations with compact leaves. From Lemmas 4.2 and 4.3 for each p ∈ C ∩M C there exists a (possibly not unique) limit curvature leaf L p ⊂ C ∩ M C which passes through p. This limit curvature leaf might degenerate to a single point.
Assume now that V ε (C) does not contain a saturated neighborhood. Then we can find a sequence of pairs of points p i , q i , each pair belonging to the same curvature leaf in M C such that p i → p ∈ C but q i ∈ V ε (C). If q is an accumulation point of q i then it cannot belong to C since |H(q)−N (q)| ε. The curvature leaf L q passing through q is contained therefore in M C. This implies that the compact curvature leaves L qi passing through q i converge in the Hausdorff topology towards L q . On the other hand we have L pi = L qi so that it also converges towards some L p . This contradicts the fact that L q is complete and thus it cannot contain points from C.
Eventually, if there does not exist some δ > 0 such that W ε ⊃ V δ (C), we would find a sequence p n ∈ V 1/n (C) W ε . If p is a limit point of the sequence p n then p ∈ W ε , as W ε is open. On the other hand p ∈ C, which is a contradiction. Now W ε is a manifold containing C and whose frontier W ε W ε ⊂ M C. Since W ε is a saturated subset, its frontier consists of union of curvature leaves. From Lemma 4.2 these are round spheres, and in particular W ε is a codimension zero submanifold of M .
Denote by N i (ε) the connected components of W ε . There are manifolds with boundary, which we call standard neighborhoods of isotropic points. The complementary M W ε is a union of spherical cylinders, which will be called necks.
The first main result is the following chopping result:  Lemma 4.8. -Assume that the compact n-manifold X can be isometrically immersed into H n+1 κ so that its boundary ∂X goes onto the round sphere S λ . Then the volume of X is greater than or equal to the volume of the standard n-ball B λ with boundary S λ lying in an n-plane H n κ embedded in H n+1 κ .
Proof. -As the orthogonal projection H n+1 κ → H n κ is distance-nonincreasing, the measure of X is greater than the measure of its projection. Now, the image of X is a homological n-cycle with boundary S λ . Then, this n-cycle cannot miss a point of B λ because the inclusion of S λ into H n κ − {pt} is a homology isomorphism.
In particular, Lemma 4.8 shows that the n-dimensional Lebesgue measure of X is uniformly bounded from below by the volume of B λ . Recall that a round sphere S λ is a metric hypersphere in a totally geodesic H n κ ⊂ H n+1 κ . Now a hypersphere of radius d in H n κ , with κ = 0, has principal curvatures √ κ/tanh( √ κd) and hence sectional curvature: Thus its volume of B λ is a function C 1 (λ, κ, n) > 0 decreasing as a function of the intrinsic curvature λ for all κ 0, namely such that: 4.6. Compactness of N i (ε). -We first prove: Proof. -Such a component N i (ε) is compact, unless its boundary ∂N i (ε) consists of infinitely many (n − 1)-spheres. All but finitely many such (n − 1)-spheres should be separating, by Grushko's theorem since each non-separating (n−1)-sphere contributes a free factor Z to π 1 (M ). Now, let M j be the compact connected submanifolds of M bounded by the separating (n − 1)-spheres in ∂N i (ε). By van Kampen's theorem and Grushko's theorem all but finitely many M j should be simply connected. Since M j is simply connected, the inclusion into M lifts to an embedding of M j into M , and thus M j is isometrically immersed into H n+1 κ such that its boundary goes onto some S λ . Now, each ∂M j is a curvature leaf isometrically embedded into H n+1 κ as a round (n − 1)-sphere S λ of curvature λ. From lemma 4.8 the n-dimensional measure of M j is uniformly bounded from below by C 1 (λ, κ, n) > 0.
Since sup p∈∂W λ λ max (ε) < ∞ by Lemma 4.7 the n-dimensional measure of M j is uniformly bounded from below by C 1 (λ max (ε), κ, n) > 0. Since M has finite volume there are only finitely many pairwise disjoint domains M j and hence only finitely many components of ∂N i (ε).   Proof. -Since N i (ε) ⊂ M is compact and simply connected it can be isometrically immersed into H n+1 κ . Moreover, the boundary ∂N i (ε) consists of finitely many (n−1)spheres. Since N i (ε) is compact and any ball in H n+1 κ is conformally equivalent to a ball in the Euclidean space R n+1 it follows that N i (ε) can be conformally immersed into R n+1 . There is then a standard procedure of gluing balls along the boundary within the realm of conformally flat manifolds (see [46, §2]). We obtain then a conformally flat closed manifold N i (ε) which is simply connected. Kuiper's theorem ( [35]) implies that N i (ε) is diffeomorphic to a sphere, and hence the lemma. Proof. -We already saw above that there are finitely many components N i (ε) with one boundary component or not simply connected. It remains to show that we cannot have infinitely many holed spheres N i (ε) with at least three holes. Assume the contrary. Then all but finitely many of them should be connected among themselves using cylinders. But a trivalent graph has at least that many free generators of its fundamental group as vertices. This implies that π 1 (M ) has a free factor of arbitrarily large number of generators, contradicting the Grushko theorem.
The only possibility left is to have infinitely many manifolds N i (ε) which are diffeomorphic to cylinders S n−1 × [0, 1], to be called tubes. Only finitely many of these tubes can be non-separating. We say that two separating tubes N i (ε) and N j (ε) are equivalent if the component of M (N i (ε) ∪ N j (ε)) joining the boundary components of the tubes is a cylinder, i.e., the tubes are isotopic in M .
A separating tube is inessential if one of the two components of its complement is a ball and essential otherwise. There are also only finitely many equivalence classes of separating essential tubes N i (ε).
In an infinite family of essential pairwise equivalent N i (ε) we have necks C i connecting them in a chain. Their union is then an open spherical cylinder. By Lemma 4.2 its closure is a closed spherical cylinder or a ball. Moreover, every N i (ε) contains a round sphere X i ⊂ N i (ε) obtained as a limit curvature leaf. Then, the limit X ∞ of X i exists and is a round sphere possibly degenerate, by the arguments from the proof of Lemma 4.3 and [8]. On the other hand X ∞ ⊂ C as H(p) = N (p) for all p ∈ X i . Thus X ∞ ⊂ W ε and, according to Lemma 4.4, sufficiently closed X i are also contained in W ε . This contradicts the fact that there is a neck C i separating X i from X i+1 which is disjoint from W ε .
If we have an infinite family of inessential tubes N i (ε) accumulating to a point q of C, then all but finitely many will be contained within the neighborhood N j (ε) containing q, leading again to a contradiction. This proves Proposition 4.5.  Proof. -We have |N (p) − H(p)| ε, for any p ∈ N i (ε). As H was supposed to be positive on M there exists some K such that min(H(p), N (p)) K > 0 for any p ∈ N i (ε).
Consider a connected component V i of π −1 (N i (ε)) , where π : M → M is the universal covering projection. Since the sectional curvature of V i is bounded from below by K > 0, the Bonnet-Myers theorem implies that V i has bounded diameter and hence it is compact. Each boundary component of V i has constant positive curvature and hence is diffeomorphic to a sphere. This implies that π 1 (V i ) is a free factor of π 1 ( M ) = 1 and so V i must be simply connected. Then the conformal gluing of balls described in [46, §2] along the boundary components of V i produces a closed simply connected conformally flat manifold V i which can be immersed into R n+1 . Therefore by Kuiper's theorem ( [35]) V i is conformally equivalent to a sphere. We derive that V i is conformally equivalent to a holed sphere, since it is obtained from a sphere by deleting out the interiors of several disjoint round balls. Now, notice that any closed loop in N i (ε) can be lifted to a path in V i , so that the action of the subgroup π 1 (N i (ε)) of π 1 (M ) as a deck transformation subgroup acting on M keeps V i globally invariant. Thus V i is endowed with a free action by π 1 (N i (ε)). Since V i is compact π 1 (N i (ε)) must be finite.
The action of π 1 (N i (ε)) on the holed sphere V i is by conformal diffeomorphisms. Then Liouville's theorem (see [38,Th. 3.1]) implies that each one of these diffeomorphisms is the restriction of some global conformal diffeomorphism of the round sphere S n , as n 3. The group generated by the global conformal diffeomorphisms of S n is still isomorphic to π 1 (N i (ε)), as any conformal diffeomorphism which is identity on an open subset of S n is the identity. The spherical metric induces then a spherical space form structure on N i (ε). Moreover, N i (ε) is obtained from the quotient of S n by the finite group π 1 (N i (ε)) by removing several open balls.
Remark 4.14. -Observe that the immersability property and the strict positive curvature enables us to find an elementary proof disposing of the differentiable sphere theorem under a pointwise pinched curvature assumption.
Remark 4.15. -Note that the π 1 (N i (ε))-action on S n might possibly not be free. Its point stabilizers might be finite, in which case the quotient S n /π 1 (N i (ε)) is a so-called orbifold. The proof actually shows that M is conformally equivalent to a connected sum of spherical space forms and classical Schottky manifolds (see [46]).
Remark 4.16. -The proof of Theorem 1.6 shows that closed manifolds having positive horizontal curvature for some 1-QC analytic metric should be conformally equivalent to either the n-sphere, S 1 × S n−1 or S 1 × −1 S n−1 . Here S 1 × −1 S n−1 denotes the non-orientable S n−1 -bundle over S 1 with structure group O(n), i.e., obtained when the monodromy map is a reflection in a hyperplane.

4.9.
Proof of Theorem 1.8. -The more general situation when H| C > 0 is treated along the same lines as before. A small neighborhood of isotropic points has boundary consisting of spheres. We wish to glue along each boundary component a ball by identifying the corresponding boundary spheres by means of some (gluing) diffeomorphism. This operation will be called capping off boundary spheres. The homeomorphism type of the closed manifold obtained by capping off its boundary components is welldefined, namely independent on the gluing diffeomorphisms. We should stress that, in contrast, the diffeomorphism type of the resulting manifold is determined by the classes of the gluing diffeomorphisms in the group of sphere diffeomorphisms up to isotopy, which can be non-trivial -for instance when n = 7. However, if the boundary spheres are endowed with Riemannian metrics of constant curvature metrics and the gluing diffeomorphism is an isometry, then the capped off manifold is well-defined up to diffeomorphism, since orientation preserving isometries are isotopic.
Our aim is to realize a geometric capping off using spherical caps of constant curvature as balls and isometries as gluing diffeomorphisms. Both the manifold and the spherical caps will be isometrically immersed in H n+1 κ . The resulting manifold inherits a C 1 Riemannian metric structure, which is C ∞ outside the joint spheres. A slight perturbation of the metric will produce a C ∞ -metric.
Our key result below shows that small neighborhoods of isotropic points are standard. We postpone the rather involved proof until the next Section 4.10.
End of proof of Theorem 1.8. -By hypothesis λ > 0 on M C and H| C > 0. From Lemma 4.6, λ has a positive lower bound on M C and by Proposition 4.5 there are only finitely many N i (ε) and finitely many necks, which are spherical cylinders. By choosing ε small enough we can assume that all N i (ε) have positive sectional curvature, because N | C = H| C > 0 and N i (ε) are saturated by Lemma 4.4. Proposition 4.17 above shows that N i (ε) are diffeomorphic to spherical space forms with several disjoint open balls removed. Thus M can be obtained by gluing together the holed spherical space forms and spherical cylinders along their boundary spheres, which correspond to attaching 1-handles. ν, for any p ∈ N i (ε), meaning that N i (ε) has pointwise pinched curvature.
Consider the manifold N i (ε) obtained topologically from N i (ε) by capping off boundary spheres by balls. We will show that N i (ε) also admits a pointwise pinched Riemannian metric.
Let S be a boundary component of N i (ε) and F be a collar of S in N i (ε) which is saturated by curvature leaves. In particular, F is diffeomorphic to S n−1 × [0, 1]. As F is simply connected, the inclusion F ⊂ M could be lifted to M and thus F is embedded into H n+1 κ . We say that F has width w if the distance between the two boundary components is w.
Before we proceed, recall that the envelope of a one-parameter family of hypersurfaces in H n+1 κ is a hypersurface tangent to each member of the family (see [19] for the case of surfaces). Alternatively, it is the boundary of the region filled by the family of hypersurfaces. If the family is given by equations F t (x) = 0, depending smoothly on the parameter t, then its envelope is the set of points x for which there exists a such that F a (x) = ∂F t /∂t| t=a (x) = 0. The envelope of a one-parameter family of hyperspheres is usually called a canal hypersurface in the literature.
We will first need the following lemma, whose proof is postponed to Section 4.11 below: (1) Assume H(S) > 0. If the width w of the collar F is small enough then we can cap off smoothly F along the boundary component S by attaching a hyperspherical cap H + (S) using an isometry between its boundary ∂H + (S) and S and of radius We suppose from now on that H(S) > 0. Let now N i (ε) denote the manifold obtained from N i (ε) by adding to each boundary component S the corresponding spherical cap H + (S). Its diffeomorphism type is uniquely determined, independently on the choice of the orientation preserving isometries used for gluing.
Note that both F and H + (S) are embedded in H n+1 κ , because they are simply connected. By choosing a collar F of small enough width we can assume that their union F ∪ H + (S) remains embedded in H n+1 κ . We actually only need this union to be immersed in H n+1 κ . Then, the Riemannian metrics on F and H + (S) are restrictions of the metric induced on their union by the embedding F ∪ S H + (S) → H n+1 κ .
We define a tensor on N i (ε) to be the 1-QC Riemannian metric on points of N i (ε) and the Riemannian metric on F ∪ S H + (S) which is induced by its embedding in H n+1 κ . Since these two metrics coincide on F this tensor is a Riemannian metric on N i (ε).
The regularity of the metric depends on the regularity of the embedding Thus the metric is C ∞ everywhere except possibly at the joint spheres ∂N i (ε) where the metric might be only of class C 1 , because F ∪ S H + (S) is only a C 1 -submanifold of H n+1 κ , in general. However, an arbitrarily small perturbation of the embedding (or the metric) by an infinitely flat function along a small collar of the boundary sphere S will make it smooth. Recall that an infinitely flat function is a smooth real function which vanishes on (−∞, 0] and equals 1 on [δ, ∞).
Further, we need to introduce another metric invariant of a Riemannian manifold. Recall from [44] that the pointwise anisotropy of the metric is the quantity: where K p (σ) is the curvature of the plane σ at p and R(p) is the scalar curvature at p. Furthermore, the integral anisotropy of the manifold N is the number N I(p) d vol .
The proof of Lemma 4.19 is deferred to Section 4.12. The last ingredient needed is the following result of Nikolaev (see [44]): Proposition 4.20. -Given any constants C 1 , C 2 there exists some δ(C 1 , C 2 , n) > 0 such that any closed n-manifold N satisfying vol(N ) C 1 > 0, diam 2 (N ) sup p∈N,σ |K p (σ)| < C 2 , and whose integral isotropy is smaller than δ, is diffeomorphic to a space form.

4.11.
Proof of Lemma 4.18. -The image of the collar F in H n+1 κ by the immersion is a canal hypersurface. This was already proved in [46,22] for the case where κ = 0.
Assume now that κ > 0. We want to show that each curvature leaf S the hypersurface F is tangent to a fixed round hypersphere H (S). Let ψ be the unitary normal vector field on the image of F in H n+1 κ and ∇ be the Levi-Civita connection on H n+1 κ . The Weingarten (or shape) operator L ψ is defined as L ψ X = (∇ X ψ) T , where the superscript T means the tangent part. Moreover, g(L ψ X, Y ) = h(X, Y ), where X, Y are tangent to M and h is the second fundamental form of M . It follows that L ψ X = − √ H + κ · X for any vector field X tangent to the curvature leaf S. shows that the measure of the spherical cap H + (S) is at least that of the hyperbolic ball B λ lying in a n-plane and having the same boundary sphere S, and thus uniformly bounded from below by the positive constant C 1 (λ max (ε), κ, n). Therefore, the volume of N i (ε) is uniformly bounded from below by C 1 (λ max (ε), κ, n) > 0.
Each boundary component of N i (ε) is either separating M in two disjoint components or non-separating. The separating spheres are of two types: either one separated component is simply connected, or else none of the components are simply connected. The boundary component is accordingly called trivial and respectively non-trivial.
Each non-separating or separating non-trivial boundary component induces a nontrivial splitting of the fundamental group π 1 (M ) as a free amalgamated product. It follows that their total number is bounded from above by the maximal number r(π 1 (M )) of factors in a free splitting of the fundamental group π 1 (M ). Further, according to Remark 4.10, the number of trivial separating boundary components is uniformly bounded by vol(M )/C 1 (λ max (ε), κ, n). Moreover, each spherical cap H + (S) lies on a hypersphere of radius d, where d is given in Lemma 4.18. It follows that the diameter of N i (ε) is uniformly bounded from above by C 2 = C 2 (vol(M ), λ, H, κ, n) = r(π 1 M ) · diam(M ) + 2πd · vol(M ) C 1 (λ max (ε), κ, n) , while the volume N i (ε) is at least C 1 = C 1 (λ max (ε), κ, n). Observe that d is uniformly bounded in terms of | inf p∈M H|. Remark 4.23. -We don't know whether pieces N i (ε) for small enough ε can be assigned a well-defined sign according to the curvature at their isotropic points. This would permit to use Gromov's pointwise pinching result in negative curvature (see [25]) under bounded volume assumptions. This would be so if locally the dimension of C were at least 3, by F. Schur ([50]).
Remark 4.24. -Isotropic points where the sectional curvature is negative can only be approached by leaves S with H(S) < 0 and the method considered above cannot provide an isometric capping off for the boundary components. For instance, if some negatively curved Gromov-Thurston manifold (see [26]) admits a spine which can be realized by points of constant sectional curvature, then we could provide examples of 1-QC manifolds which are not obtained by gluing together space forms. Recall that a spine for a closed manifold M is a polyhedron P ⊂ M with the property that M D n collapses onto P , where D n ⊂ M denotes an n-ball. A typical spine could be geometrically constructed using the cut locus of nice metrics.

5.
More about the equivariant immersions of 1-QC manifolds Throughout this section M is a closed orientable n-manifold endowed with a 1-QC Riemannian metric, which is assumed conformally flat when n = 3. Remark 5.7. -When n = 2 the situation is different, and it was described by Labourie in [41]. The space I (k 0 ) of codimension-one isometric equivariant isometries of a surface of a surface S of genus at least 2 into the hyperbolic space H 3 1 , such that the pull-back metric is of constant curvature k 0 ∈ (−1, 0) -up to left composition by isometries of H 3 1 and right composition by lifts of diffeomorphisms isotopic to identity -is not one point as for n 4. In fact, there is a natural map of I (k 0 ) into the Teichmüller space T (S) which associates to an immersion the conformal class ι 0 of the induced constant curvature metric. The fiber I (k 0 , ι 0 ) of this map can also be mapped homeomorphically onto the Teichmüller space T (S) by sending the class of an immersion to the conformal class of its second (or third) fundamental form or, equivalently, of its associated complex structure. Then the map from I (k 0 ) to the space of CP 1 (i.e., conformally flat) structures on S which associates to an immersion f the CP 1 -structure given by F is also a homeomorphism. This gives a one parameter family of parametrizations of the space of CP 1 structures by T (S)×T (S).
Thus, for each CP 1 -structure on S and k ∈ (−1, 0) there exists some metric of constant curvature k on S and an equivariant isometric embedding f k : S → H 3 1 , whose associated CP 1 structure is the one with which we started. Further, the immersion φ k associated to f k provides a hyperbolic structure M (k) on S × [0, ∞), called a geometrically finite end (see [41]). In [41,40] one proved that for each constant k ∈ [k 0 , 0) there exists a unique incompressible embedded surface S k ⊂ S × [0, ∞) of constant curvature k homeomorphic to S and the family S k foliates the geometrically finite end M (k 0 ). This proves that actually M (k 1 ) ⊂ M (k 2 ) if k 1 > k 2 . When k converges to −1 then M (k) converges to a geometrically finite end. The CP 1 -structure associated to this geometrically finite end is also the one from the beginning. Moreover, the finite boundary of M (0) = k M (k) is isometric to a surface of constant curvature −1 but its embedding into H 3 1 is a pleated surface along a geodesic lamination, according to Thurston (see [41] for more details). Higher dimensional versions of Labourie's results concerning moduli of hyperbolic flat conformal were obtained by Smith in [51].