The topology of closed manifolds with quasi-constant sectional curvature

arXiv:1406.0327v1 [math.DG] 2 Jun 2014

THE TOPOLOGY OF CLOSED MANIFOLDS WITH QUASI-CONSTANT SECTIONAL CURVATURE LOUIS FUNAR Abstract. We prove that closed manifolds admitting a metric whose sectional curvature is locally quasi-constant are graphs of space forms. In the more general setting of QC space 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 space forms and spherical bundles over the circle. AMS Math. Subj.Classification: 53C21, 53C23, 53C25, 57R42.

1. Introduction The classification of (locally) conformally flat manifolds seems out of reach in full generality since M. Kapovich proved (see [21]) 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 manifold of dimension (at least) 4. On the other hand, fundamental groups of conformally flat manifolds cannot be arbitrary, for instance they cannot be simple finitely presented infinite groups (e.g. Thompson’s groups). Kamishima ([20]), improving previous work by Goldman ([13]) for virtually nilpotent groups, showed that a closed conformally flat n-manifold with virtually solvable fundamental group is conformally equivalent to a n-spherical space form, an Euclidean space-form or it is finitely covered by some Hopf manifold S 1 × S n−1 . Further Goldman and Kamishima classified up to a finite covering in [14] the conformally flat closed manifolds whose universal covering admits a complete conformal Killing vector field, by adding to the previous list the closed hyperbolic manifolds, their products with a circle and conformally homogeneous quotients of the form (S n − lim Γ)/Γ, where lim Γ ⊂ S n−2 is the limit set of the conformal holonomy group Γ. The aim of this paper is to describe the topology of a family of conformally flat manifolds admitting a local vector field with respect to which the curvature is quasi-constant. This hypothesis is similar in spirit to the one used by Goldman and Kamishima ([14]), but in our case the vector field is not a conformal Killing field, in general, and the methods of the proof are rather different coming from foliations and Cheeger-Gromov’s theory. The spaces under consideration are closed manifolds admitting a Riemannian metric of quasi-constant sectional curvature of dimension n ≥ 3. These spaces were first studied 1

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from a geometrical point of view by Boju and Popescu in [2] and coincide with the kspecial conformally flat spaces considered earlier in a different context by B.-Y.Chen and K.Yano ([17, 8]). These were further analyzed by Ganchev and Mihova in [12]. It was noticed in [1, 17] and later in [12] that these are locally conformally flat manifolds for n ≥ 4. The original definition of [2] (see also the comments of the last section in [2]) is too restrictive for our purposes and we will call it global in the sequel, as follows: Definition 1.1. The (metric of the) Riemannian manifold M is globally 1-QC with respect to a non-vanishing vector field ξ, called distinguished vector field, if the sectional curvature function is constant along the cone of 2-planes making a constant angle θ(p) ∈ (0, π2 ) with ξ, in every point p ∈ M. Furthermore the metric is globally 1-QC if there exists some non-vanishing vector field ξ and a smooth angle function θ as above. The topology of globally 1-QC manifolds is rather simple, as we will see that they must be cylinders. Further the geometry of globally 1-QC manifolds was completely described under an additional condition (namely, that the distinguished field be geodesic) in ([12], section 6): these are precisely the warped product Riemannian manifolds between some closed space form and an interval (see also section 6.1 below), and also appeared as the so-called subprojective spaces. Observe that the definition makes sense also when ξ is a field of directions. It is proved in [2] that then the sectional curvature function is constant along any cone of 2-planes making a constant angle in [0, π2 ] with ξ, in every point. This pointwise condition depends on the choice of a non-vanishing vector field, which imposes unnecessary topological obstructions. We slightly weaken it as follows: Definition 1.2. The (metric of the) Riemannian manifold M is called (1) local 1-QC, if any point p has an open neighborhood which is globally 1-QC with respect to the induced metric. (2) 1-QC if any non-isotropic point for the sectional curvature function has an open neighborhood which is locally 1-QC. Recall that a point is called isotropic if all tangent planes have the same sectional curvature. Observe that a local 1-QC manifold is a 1-QC manifold. The sectional curvature N and H of 2-planes containing and respectively orthogonal to the vector field ξ will be called vertical and horizontal curvature functions, respectively. It will become clear in the next section that both H and N extend smoothly on all of M. Definition 1.3. 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 cylinders connecting pairs of boundary components. The pieces will be accordingly called vertex manifolds and edge manifolds or tubes. Our first result is a rather general description of the topology of the local 1-QC spaces: Theorem 1.1. Let M n be a closed local 1-QC n-manifold, n ≥ 3, which we assume conformally flat, when n = 3. Then M is diffeomorphic to a graph of space forms.

TOPOLOGY OF QC SPACES

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Our second result aims to extend this to the case of 1-QC manifolds. Theorem 1.2. Let M n be a closed n-manifold, n ≥ 3, admitting a 1-QC metric. When n = 3 we also assume that M 3 is conformally flat. (1) If π1 (M) is finite then M is conformally equivalent to a spherical space form. (2) If π1 (M) is infinite, we assume that it is torsion-free. If the horizontal curvature H is positive, then M is conformally equivalent to a connected sum of spheres which can be realized by adjoining spherical cylinders along totally umbilical embedded codimensionone spheres. In particular, M is diffeomorphic to an n-sphere with several (possibly unoriented) 1-handles. Recall that a n-sphere with a number of (possibly unoriented) 1-handles added is a connected sum ♯k S 1 × S n−1 or ♯k S 1 ×−1 S n−1 , where S 1 ×−1 S n−1 denotes the (generalized) Klein bottle fibering on the circle and having monodromy map of degree −1, so that the manifold is non-orientable. Remark 1.1. If both H and N, and thus the Ricci curvature, are positive, classical results of Bochner, Lichnerowicz and Myers show that the manifold is a homology sphere with finite fundamental group. In particular manifolds with 1-handles in the second part of the theorem cannot have metrics with positive N. Assume now that M has not necessarily positive horizontal curvature. Nevertheless there exists another quantity, called the curvature leaves functions λ, whose positivity has also strong consequences for the topology of leaves. Specifically, on all non-isotropic points we set (see also the section 2.4): λ=H+

k gradH k2 4(H − N)2

Theorem 1.3. Let M n , n ≥ 3, be a closed n-manifold with infinite torsion-free π1 (M), admitting a 1-QC metric, which is conformally flat when n = 3. Moreover, suppose that there are no isotropic points having non-positive sectional curvature. Then M is diffeomorphic to an n-sphere with several (possibly unoriented) 1-handles. Remark 1.2. It seems that whenever π1 (M) is one-ended (possibly with torsion) then M is a spherical space form. However, the methods of this paper do not suffice to show this for n ≥ 4. All over this paper (local) 1-QC manifold or space will mean a manifold which admits a Riemannian (local) 1-QC metric. Acknowledgements. The author is indebted to Gerard Besson, Zindine Djadli, Bill Goldman, Vlad Sergiescu and Ser Peow Tan for useful discussions. The author was partially supported by the ANR 2011 BS 01 020 01 ModGroup and he thanks the Erwin Schrödinger Institute for hospitality and support.

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2. Preliminaries 2.1. Non-isotropic points of local 1-QC manifolds. Any p ∈ M has an open neighborhood U containing and a vector field ξU , nonzero on U such that the metric is globally 1-QC on U relative to ξU . If U ∩ V 6= ∅ then let CU V = {p ∈ U ∩ V ; ξU |p 6= ±ξV |p } Then, by linear algebra we find that the points of CU V should be isotropic for the sectional curvature K. If C is the union of various CU V , where U, V run over a finite covering of the compact M then C is closed. Moreover M − C is open and admits a field of directions (in general not of vectors) everywhere nonzero with respect to which the metric is globally 1-QC. A double covering of it has a non-zero vector field and the induced metric is globally 1-QC. We obtained then: Lemma 2.1. A 1-QC manifold M, in particular a local 1-QC manifold, is the union of an open subset M − C which is globally 1-QC and a closed subset C of isotropic points. Using the field of directions ξ on M − C we obtain the smooth functions H and N of horizontal and vertical curvatures, which are defined on M − C. We extend them over the set of isotropic functions by setting H(p) = N(p) = K(p) where K is the sectional curvature at p (which is independent on the choice of the 2-plane). It is clear that H and N are still smooth functions on M. Observe that any open set of isotropic points is actually of constant curvature, by F. Schur’s theorem ([36]). Eventually we define the horizontal distribution D to be the orthogonal to the distinguished line field ξ; this is well-defined on M − C. Remark 2.1. The meaning of the 1-QC condition is that we have a global 1-QC structure for which the distinguished vector field is allowed to vanishes on the set of isotropic points, while the local 1-QC condition means that there exist non-zero extensions around each point of M. One might consider some other conditions of 1-QC type, by allowing for instance the distinguished local vector fields ξU to have also isolated singularities at nonisotropic points. It seems however that the set of diffeomorphism types of 1-QC closed manifolds in this broader sense is not larger than that obtained with our former definition. For simplicity of the exposition we assume from now on that ξ is a vector field throughout the paper, unless the opposite is explicitly stated. When ξ is a line field one should pass to a double cover. When M is local 1-QC we can assume without loss of generality that ξU are unit vector fields. When M is 1-QC it is often more useful to think at ξ as a smooth vector field on M vanishing at C which is unit outside a small neighborhood of C.


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2.2. Conformal flatness. A locally conformally flat space has a Riemannian metric which is locally conformal to the Euclidean metric. We will drop the adjective local from now on in the sequel. It is well-known that all rational Pontryagin forms, and hence classes, vanish on conformally flat manifolds (see [9, 25]. Moreover there are not known any other topological obstructions beyond the fundamental group, to the existence of conformally flat structures (see also [21]). Conversely, it seems unknown whether conformally flat manifolds admit conformally flat metrics for which the curvature tensor in suitable basis has all its components with 3 and 4 distinct indices vanishing. An instructive example of this type are the spaces of quasi-constant curvature. One important ingredient used in this paper is the following easy observation (see also [1, 12]): Lemma 2.2. (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. Proof. This is already known when the dimension n ≥ 4 because the Weyl tensor vanishes (see [1, 12]). Since the Weyl tensor vanishes identically in dimension 3 the Schouten tensor S is determined uniquely by the following identity (see [28]): R(X, Y, Z, T ) = g(Y, Z)S(X, T ) − g(X, Z)S(Y, T ) + g(X, T )S(Y, Z) − g(Y, T )S(X, Z)

We find then

H g(X, Y ) + (N − H)g(X, ξ)g(Y, ξ) 2 By the Weyl-Schouten theorem (see [28], section C, Thm. 9) the 3-manifold is conformally flat iff the so called Cotton-Schouten tensor vanishes, namely if: S(X, Y ) =

(∇X S)(Y, Z) − (∇Y S)(X, Z) = 0

Define the 1-form η by η(X) = g(ξ, X), such that the horizontal distribution is the kernel of η. The previous condition is equivalent to: dH(X)g(Y, Z) − dH(Y )g(X, Z) + 2(N − H)dη(X, Y )η(Z)+ +2(N − H) (η(Y )(∇X η)Z − η(X)(∇Y η)Z) +

+2 (d(N − H)(X)η(Y ) − d(N − H)(Y )η(X)) η(Z) = 0 Using all possibilities for choosing X, Y, Z among the elements of a basis of the horizontal distribution D or ξ it follows that this is equivalent to the system of the following three equations: dH = ξ(H)η; ξ(H) ∇X ξ = X, if X ∈ D; 2(N − H) dη(X, Y ) = 0, if X, Y ∈ D. The first two equations are always verified by using the method of [2] in all dimensions n ≥ 3. The third one is equivalent to the fact that the horizontal distribution D is completely integrable.

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Remark 2.2. One cannot dispose of the integrability condition 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 = − 14 and N = − 34 . 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 [13]) unless they are finitely covered by a sphere, a torus or S 1 × S 2 . 2.3. Isometric immersions. One important tool which will be used throughout this paper is the codimension one immersability of simply connected 1-QC manifolds. Let Hm 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 1 − inf p∈M H(p), when inf p∈M H(p) ≤ 0; κ= 0, when inf p∈M H(p) > 0. The main result of this section is:

f Proposition 2.3. Assume M n admits a 1-QC metric. Then the universal covering M n+1 admits an isometric immersion into Hκ . f is a (possibly non-compact) 1-QC manifold with the Proof. The universal covering M f → M. Define the following (smooth) symmetric induced metric by the covering π : M (1,1) tensor field on M pointwise, by means of the formula: ( √ p g(ξ,V )|p −H · g(ξ,U )|kξk , if p ∈ M − C; H + κ · g(U, V )|p + √NH+κ 2 √ h(U, V )|p = H + κ · g(U, V )|p , if p ∈ C,

f One verifies immediately the Gauss and Codazziand also denote by h the pull-back to M. Mainardi equations associated to the tensor h, namely:

R(X, Y, Z, W ) = −κ (g(Y, Z)g(X, W ) − g(X, Z)g(Y, W ))+h(Y, Z)h(X, W )−h(X, Z)h(Y, W ) ∇X h(Y, Z) = ∇Y h(X, Z)

This follows from the second Bianchi identity of the curvature tensor, when n ≥ 4 and from the vanishing of the Schouten tensor in dimension 3. The second equality is treated as in the proof of lemma 2.2 by noticing that only the second condition discussed there has to be verified, the other ones being automatic. Therefore, by the fundamental theorem for submanifolds of constant curvature space, (see [38], chap.7, section C, Thm. 21) as f is a connected simply connected n-manifold then it admits an isometric immersion M f : M → Hn+1 κ , whose second fundamental form is h.


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Remark 2.3. The immersability result holds also when M n is not necessarily compact, provided inf p∈M H(p) exists. 2.4. The leaf curvature function λ. It is proved in [2] that the distribution D is completely integrable for n ≥ 4, while for n = 3 this is part of our assumptions, and thus D defines a foliation of M − C. The leaves of the foliation will be called curvature leaves. According to [2] every curvature leaf is a totally umbilical submanifold of M of intrinsic constant curvature equal to: ξ(H)2 λ=H+ 4(N − H)2 Lemma 2.4. If M is a local 1-QC manifold then there is a smooth extension of λ to M − C.

Proof. If ξU is the distinguished local vector field on U where U ∩ C 6= ∅ and X is orthogonal to ξU and part of a holonomic basis of the tangent bundle then we have (see [2, 12]): ξU (H) = g(∇X ξU , X) 2(N − H) We put therefore: λ(p), if p ∈ U − C; λU (p) = H + (g(∇X ξU , X))2, if p ∈ U ∩ C ∩ M − C, where X is orthogonal to ξU . Notice that ξU is well-defined on M − C. Moreover, the extensions λU are pairwise compatible, namely if p ∈ U ∩ V ∩ C then λU (p) = λV (p). We denote therefore by λ this extension to a smooth function on all of M − C. We also denote by α the quantity g(∇X ξU , X) in the sequel, so that curvature leaves are totally umbilical with principal curvatures equal to α. Remark 2.4. The meaning of this lemma is that λ remains bounded when it approaches C. In particular, we cannot have isolated singularities of the vector field ξ having a ball neighborhood foliated by constant curvature concentric spheres. Remark 2.5. Recall also (see [2, 12]) that |ξ(H)| =k grad H k

and the function λ is constant on the level hypersurfaces of H. 3. Proof of Theorem 1.1 3.1. Globally 1-QC components are cylinders. The purpose of this section is to prove that the globally 1-QC manifolds have a rather simple topological structure: Proposition 3.1. Let E be a connected component of M − C. If E is compact then it is diffeomorphic to a fibration over S 1 . If E is non-compact then E is diffeomorphic to N × R, where N is a closed space form.

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Corollary 3.1. If E is a connected component of M − C then λ(p) is either positive, or negative, or else vanishes for all p ∈ E. Before to proceed we will need the following: Proposition 3.2. There exists another foliation of E, called the Novikov simplification of the foliation, whose leaves have the same intrinsic curvature function λ and moreover, all leaves are compact. Proof of proposition 3.1 assuming proposition 3.2. Suppose that there exist p+ , p− ∈ E for which the associated leaves are not diffeomorphic. Consider a smooth generic path in E joining the the leaves L+ and L− . The union of all leaves intersecting the path is a compact connected codimension zero submanifold of E which is foliated by the curvature leaves and whose boundary contains both leaves L± . The strong form of the global Reeb stability theorem states that a codimension one transversely orientable foliation whose all leaves are compact is locally a fibration and the holonomy groups are finite (see [29, 11]). 3.2. Novikov components and the proof of proposition 3.2. We recall now the definition of Novikov components of a foliation (see [31]): Definition 3.1. Two points of the 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 main ingredient needed in this section is the following classical result due to Novikov: Lemma 3.3 ([31]). Every Novikov component is either a compact leaf which does not admit any closed transversal or else an open codimension zero submanifold whose closure has finitely many boundary components consisting of compact leaves. Proof of proposition 3.2. The function H is smooth and proper, because M is compact. Then critical values are nowhere dense and the preimage of any non-critical value is a codimension one compact submanifold of M. A curvature leaf L will be called non-critical if H(L) is a non-critical value. Lemma 3.4. A non-critical curvature leave L is a connected component of H −1 (H(L)), and in particular it is compact. f is complete, following Reckziegel Proof. Each component of the preimage π −1 (L) of L in M ([34]). 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 is of infinite degree. In particular L is not necessarily a complete subspace of M and in particular it might not to be a closed subset of M if the inclusion of L into M is not proper.


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If H(L) is a non-critical value of H then L is a codimension zero submanifold of the manifold H −1 (H(L)). The level hypersurface H −1 (H(L)) 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. When H(L) = c0 is a critical value of H there exist some p ∈ L such that d(H)|p = 0. Now, p ∈ M − C and dH = ξ(H)η, thus ∇X ξ = αX = Nξ(H) X vanishes at p, for any −H horizontal vector field X. In particular α(p) = 0 and λ(p) = H(p). Since λ is constant on the level hypersurfaces of H it follows that λ(q) = H(q) = c0 , for any q ∈ L and thus the curvature leaf L is totally geodesic. Assume for the moment that H is not globally constant on E. Therefore there exists non-critical values c± with c− < c0 < c+ , with |c+ − c− | arbitrarily small. Therefore H −1 ([c− , c+ ]) is a compact submanifold with boundary foliated by the curvature leaves, each boundary component being a leaf. Suppose now that the critical leaf L is not compact. It follows that L is contained into some Novikov component U, whose closure U ⊂ H −1 ([c− , c+ ]) is compact. Observe that a Novikov component is a saturated closed subset. Furthermore U cannot contain any compact leaf and thus all leaves contained in U must be non-compact. Since every leaf in U approaches some boundary leaf it follows also that H must be constant on U. Moreover, from above all leaves in U are totally geodesic. We claim now that: Proposition 3.5. The manifold U is a cylinder. Proof. Each component of ∂U has a sign determined by the fact that ξ points either inward or outward with respect to U. This is well-defined because ξ is a smooth vector field on M orthogonal to ∂U , so that if ξ points inward at some point of ∂U then it points inward at all points of its connected boundary component. This means that the associated foliations is transversely orientable. Let ∂ + U be the union of those components of ∂U for which ξ is inward pointing and ∂ − U be its complementary. Observe that U cannot be a closed manifold without boundary since it is contained in E. By the symmetry of the situation we can assume that ∂ + U is non-empty. We claim now that: Lemma 3.6. Every integral trajectory γx of ξ starting at a point x ∈ ∂ + U should cross ∂−U . Proof. Assume the contrary. As γx cannot cross again ∂ + U , necessarily in outward direction, because this would contradict the choice of ∂ + U , γx will remain forever in U. Thus its ω-limit set ω(γx ) is a compact invariant subset of U . If ω(γx) ∩ ∂U 6= ∅, then γx will contain points y arbitrarily closed to a curvature leaf for which γ˙x |y = ξ|y is very closed to the horizontal distribution, which contradicts the fact that the horizontal distribution is everywhere orthogonal to ξ. Henceforth ω(γx ) ∩ ∂U = ∅. Let then L be some boundary leaf, component of ∂U and β : [0, 1] → U be a geodesic in U realizing the positive distance between the compact

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ω(γx ) and boundary leaf L. Let β(0) ∈ ω(γx ). Since ω(γx ) is invariant by ξ the integral orbit O of β(0) under the one-parameter flow defined by ξ is contained in ω(γx ). Standard variational arguments show that β must be orthogonal to both the orbit O and the leaf L, namely that ˙ ˙ g(β(0), ξ|β(0) ) = 0, g(β(1), D|β(1) ) = 0 ˙ is tangent to the leaf Lβ(0) passing through β(0) ∈ U. The first equation means that β(0) But the leaf Lβ(0) cannot be compact, since it is contained in the Novikov component U. Thus Lβ(0) is a critical leaf and so it is a totally geodesic leaf. Eventually, a geodesic in U whose direction is tangent to a totally geodesic leaf must be contained in that leaf Lβ(0) . On the other hand the second equation above shows that β is orthogonal to the leaf L at β(1) which is a contradiction. This ends the proof of the lemma. Let then denote by l(x) the length of the integral trajectory γx between x and its endpoint on ∂ − U. We consider then φ : ∂ + U × [0, 1] → U defined by: t φ(x, t) = γx l(x)

This is obviously a diffeomorphism on its image. The only obstruction for the surjectivity of φ is the existence of closed trajectories of ξ. The proof of the previous lemma shows that there are no closed orbits of ξ contained in U. This proves proposition 3.5.

Now we define the Novikov simplification of a foliation as above to be result of excising every open Novikov component and gluing back an open cylinder with its standard foliation. It follows that the Novikov simplification of the curvature foliation of E must have only compact leaves, when H is not globally constant. When H is constant on all of E, then E itself is one Novikov component. We can apply the argument from above to E (see below) and find that E is a cylinder. This ends the proof of proposition 3.2 This shows that we can attach a sign to each connected component of M −C, as follows: E is positive, vanishes or is negative if λ > 0, λ = 0 or λ < 0 on E. 3.3. The limit leaves. The limit subspace of a 1-QC manifold is the compact set L = M − C ∩ C. When the manifold is local 1-QC this limit set has a simple structure, as follows: Proposition 3.7. The limit subspace L is a lamination with isolated compact leaves. In particular M − C and the connected components of C are compact manifolds with boundary. Proof. Every p ∈ L can be approached by points pi ∈ M − C. In a small open neighborhood U of p we can express ξU |p = limi→∞ ξ|pi . Since the right hand side is independent on U it follows that ξ|p = ξU |p is well-defined and independent on M − C. Therefore the horizontal distribution has a smooth involutive extension to M − C, and hence it is completely integrable. This implies that the curvature leaf Lp passing through p is


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a smooth manifold. The previous results imply that Lp is compact. Therefore L is a lamination. Since L has no interior points each connected component must be a compact leaf. Moreover, each connected component of M − C is a cylinder over some compact leaf. We say that leaves L in L are double limit leaves if they are contained in int(C)∩M − C. The closure M − C must have only finitely many connected components. However L might have infinitely many connected components, for instance an infinite union of leaves accumulating on a leaf. However these leaves are contained in int(M − C) are not double limit leaves. In fact the boundary of M − C consists of the double limit leaves. Eventually M is the union of manifolds of constant curvature int(C) and a number of cylinders M − C. Since the limit leaves are also totally umbilical and of constant curvature M is a graph of space forms. This proves theorem 1.1. 4. Proof of Theorem 1.2 4.1. Outline. The first step rely on methods of conformal geometry. Specifically, using the codimension-one isometric immersion of the universal covering permits to understand the geometry and the topology of compact simply connected subsets of non-isotropic points. In particular the positivity assumption imply that the open set of non-isotropic points consists of spherical cylinders whose sections are controlled by λ. In the second step we aim at getting finiteness statements concerning the topology of the cylinders and the saturated regular neighborhoods of the isotropic points. The main difficulty is that the set of isotropic points might a priori be topologically wild. The 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. The third step deals with the geometry of the regular neighborhoods of isotropic points. When the horizontal curvature is positive their universal coverings must be compact and Kuiper’s theorem show that their conformal geometry is spherical. When only the quantity λ is positive we cannot use this trick. However, a deep result of Nikolaev states that a manifold with bounded volume and diameter whose integral anisotropy is small enough should be diffeomorphic to a space form. We show that small width spherical cylinders in codimension one are envelopes of 1-parameter families of hyperspheres in the hyperbolic space. We can cap off boundary components by means of hypersphere caps in the hyperbolic space whose geometry is controlled. 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. Finite fundamental group. Assume that M has finite fundamental group. Then the claim follows from the following: Proposition 4.1. If M is a closed simply connected 1-QC n-manifold of dimension n ≥ 3 (conformally flat when n = 3) then it is conformally equivalent to a sphere.

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Proof. This is part of Kuiper’s theorem (see [22]) which states that whenever a compact simply-connected manifold admits a locally conformally flat C 1 metric then the metric is actually the pull-back of the round metric on the sphere under some immersion. Moreover, if the manifold is also closed then the immersion is a covering and hence the manifold is conformally equivalent to to a round sphere. Remark 4.1. Notice that there are no assumptions concerning the sign of the curvatures functions H or N above. 4.3. Positive horizontal curvature. We assume from now on that the compact 1-QC manifold M has infinite fundamental group, which in this section will be torsion-free. Proposition 4.2. When λ(p) > 0 for any p ∈ M − C, in particular when H > 0, the curvature leaves are diffeomorphic to spheres and they foliate the open set M − C of non-isotropic points. f there exists a principal curvature Proof. According to Cartan (see [4]) at each point p ∈ M µ of the immersed manifold of multiplicity at least n − 1. Let N be the open set of nonumbilical points, namely those for which µ has multiplicity exactly n − 1. An immediate computation of the eigenvalues √ of h in a local orthonormal basis+κfor , g shows that the principal curvatures are µ = H + κ of multiplicity n − 1 and √NH+κ f with respect to this isometric respectively. Thus the open set N of non-umbilic points of M f, namely the inverse image immersion coincides with the set of non-isotropic points of M of M − C to the universal covering. Furthermore the lift of the horizontal distribution D √ to N is the eigenspace of the second fundamental form h corresponding to H + κ. Moreover N is foliated by curvature leaves (see [10], Thm. 1.4). Following Reckziegel f are complete. Moreover, each ([34], see also [32], section 2) the curvature leaves in M f has constant sectional curvature equal to the pull-back of λ to M f and curvature leaf of M hence is bounded below by a positive number. By Bonnet-Myers theorem each curvature leaf has finite diameter and its completeness implies that it is compact. Therefore each leaf is mapped isometrically onto some round (n − 1)-sphere by the immersion in Hn+1 κ (see e.g. [33]). Recall that round n-spheres (sometimes also called geodesic or extrinsic n-spheres) are intersections of a hypersphere with a hyperbolic n-plane (see [33], section 2). f and thus Now any curvature leaf L in M is covered by a disjoint union of spheres in M f is finite L is diffeomorphic to a spherical space forms. The stabilizer of one sphere in M and obviously contained in π1 (M). Thus it must be trivial as π1 (M) was assumed to be torsion-free and hence the curvature leaf L is a sphere. This settles proposition 4.2. When C = ∅ then M is diffeomorphic to S 1 × S n−1 or S 1 ×−1 S n−1 . Moreover, it follows from ([32], section 2) that M is conformally equivalent to S 1 × S n−1 or S 1 ×−1 S n−1 . Suppose now that there exist isotropic points, so that C 6= ∅. From above, any compact component of M −C is also diffeomorphic to S 1 ×S n−1 or S 1 ×−1 S n−1 . Further, proposition 4.2 and the Reeb stability theorem show that each open connected component of M − C


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is diffeomorphic to a spherical cylinder S n−1 × (0, 1). The complementary in M of the union of slightly shrinked tubes S n−1 × (ǫ, 1 − ǫ) has the connected components Ni (ε). Proposition 4.3. Assume that λ(p) > 0 for p ∈ M. Then the components Ni (ε) are compact. Proof. The component Ni (ε) is compact, unless its boundary consists of infinitely many (n − 1)-spheres. Assume the contrary. All but finitely many such (n − 1)-spheres should be separating, by Grushko’s theorem since each non-separating (n − 1)-sphere contribute a free factor Z to π1 (M). Each separating (n − 1)-sphere bounds a compact connected manifold and all but finitely many of these are simply connected since their fundamental group is also a free factor of π1 (M). We saw above that each boundary component, which is a curvature leaf is isometrically embedded into Hn+1 as a round (n − 1)-sphere Sλ of curvature λ. κ Assume for the moment that λ(p) > 0 for all p ∈ M. By compactness of M and the fact that λ is continuous on all of M we know that λ is uniformly bounded both from above and away from zero. Lemma 4.4. Assume that the compact n-manifold X can be isometrically immersed into Hn+1 so that its boundary ∂X goes onto the round sphere Sλ . Then the volume of X is κ greater than or equal to the volume of the standard n-ball Bλ with boundary Sλ lying in a hyperbolic n-plane Hnκ of Hn+1 κ . Proof. This follows from the fact that the orthogonal projection of X onto this hyperbolic n-plane is contracting and its image is a homological cycle with boundary Sλ . Thus, at first the measure of X is greater than the measure of its projection. Then, by homological reasons this n-cycle should contain Bλ . In particular, the n-dimensional Lebesgue measure of X is uniformly bounded from below by some constant C1 (λ, κ, n) > 0, which is decreasing as a function of λ. Now, let Mj be the compact connected and simply connected submanifolds of M bounded by the separating (n − 1)-spheres components of Ni (ε). Since Mj is simply f, and thus Mj is isoconnected, the inclusion into M lifts to an embedding of Mj into M n+1 metrically immersed into Hκ such that its boundary goes onto some Sλ . From lemma 4.4 the n-dimensional measure of Mj is uniformly bounded from below by C1 (λ, κ, n) > 0. Since supp∈M λ < ∞ the n-dimensional measure of Mj is uniformly bounded from below by C1 (supp∈M λ, κ, n) > 0. But the set of Mj ’s consists of infinitely many pairwise disjoint domains contained in M. This contradicts the fact that M has finite volume, since it is compact. Therefore every Ni (ε) has finitely many components and thus it is compact. Remark 4.2. The proof above shows that the number of separating boundary components which separate simply connected domains is uniformly bounded by C1 (supvol(M )λ,κ,n) . p∈M

If there were infinitely many such Ni (ε) then all but finitely many would be simply connected by Grushko’s theorem, since π1 (M) is a free product of the groups π1 (Ni (ε)) and a free group, the latter being the contribution of the non-separating tubes.

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Lemma 4.5. If Ni (ε) is simply connected then it is diffeomorphic to a sphere with finitely many disjoint open disks removed. Proof. As in the previous lemma, since Ni (ε) is compact and simply connected it can be isometrically immersed into Hn+1 κ . Moreover, the boundary ∂Ni (ε) consists of finitely many (n − 1)-spheres. Since Ni (ε) is compact and any ball in Hn+1 is conformally equivκ alent to a ball in the Euclidean space Rn+1 it follows that Ni (ε) can be conformally immersed into Rn+1 . There is then a standard procedure of gluing balls along the boundary within the realm of conformally flat manifolds (see [32], section 2). We obtain then [ a conformally flat closed manifold N i (ε) which is simply connected. Kuiper’s theorem [ implies that N i (ε) is diffeomorphic to a sphere, and hence the lemma. Lemma 4.6. There are only finitely many components Ni (ε) others than cylinders. Proof. We already saw in the proof of lemma 4.3 that there are finitely many components Ni (ε) with one boundary component or not simply connected. It remains to show that we cannot have infinitely many holed spheres Ni (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 π1 (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 Ni (ε) which are diffeomorphic to cylinders S n−1 × [0, 1]. Observe again that only finitely many of these tubes can be non-separating. We call them inessential tubes. We might dispose of inessential tubes by slightly perturbing the metric, but we won’t need this in the sequel. The remaining components Ni (ε) will be called essential and there are only finitely many of such. The complementary of essential Ni (ε) are diffeomorphic to finitely many tubes. The pieces Ni (ε) correspond to cutting M open along a maximal collection of embedded curvature leaves spheres, pairwise not homotopic and not null homotopic. Before to proceed we will need the following technical lemma: Lemma 4.7. Suppose that λ > 0 on M − C. Then there exist arbitrarily small saturated neighborhoods of C, namely for each open neighborhood V of C in M there exists a neighborhood W ⊂ V such that W − C is a union of leaves. Proof. This is a classical result in codimension one foliations with compact leaves. Since the argument is simple we outline it. It is enough to consider V of the form V = {p ∈ M; |H(p) − N(p)| < ε}

Since the horizontal distribution D is involutive on M − C, its extension to the closure M − C is also involutive, by continuity. In particular, when p ∈ C ∩ M − C then there exists a curvature leaf Lp which passes through p. Assume now that V does not contain a saturated neighborhood. Then we can find a sequence of pairs pi , qi belonging to the same curvature leaf in M − C such that pi → p ∈ C but qi 6∈ V . If q is an accumulation point of qi then it cannot belong to C since


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|H(q) − N(q)| ≥ ε. The curvature leaf Lq passing through q is contained therefore in M − C. This implies that the compact curvature leaves Lqi passing through qi converge in the Hausdorff topology towards Lq . On the other hand we have Lpi = Lqi so that it also converges towards Lp . This means that Lp = Lq , contradicting the fact that Lq is complete and thus it cannot contain points from C. Actually this shows that Lp should be contained in C ∩ M − C, whenever p ∈ C ∩ M − C. Lemma 4.8. Assume that H > 0. If Ni (ε) is an essential piece then it is conformally equivalent to a spherical space form with several disjoint open disks removed. Proof. By choosing ε > 0 small enough by the saturation lemma 4.7 we can assume that |N(p) − H(p)| ≤ δ, for any p ∈ Ni (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 ∈ Ni (ε). f → M is the universal Consider a connected component Vi of π −1 (Ni (ε)) , where π : M covering projection. Since the sectional curvature of Vi is bounded from below by K > 0, the Bonnet-Myers theorem implies that Vi has bounded diameter and hence it is compact. In this situation we can apply (Theorem 1.4. (ii) from [10]) to the compact submanifold f which is immersed in Rn+1 . Remark also that ∂Vi consists only of spheres. This Vi of M f and so Vi must be simply connected. Then the implies that π1 (Vi ) is a free factor of π1 (M) conformal gluing of balls described in ([32], section 2) along the boundary components of Vi produces a closed simply connected conformally flat manifold Vbi which can be immersed into Rn+1 . Therefore by Kuiper’s theorem Vi is a holed sphere. Now, notice that any closed loop in Ni (ε) can be lifted to a path in Ni (ε), so that the action of the subgroup π1 (Ni (ε)) of π1 (M) as a deck transformation subgroup acting on f keeps Vi globally invariant. Thus Vi is endowed with a free action by π1 (Ni (ε)). Since M Vi is compact π1 (Ni (ε)) must be finite. The action of π1 (Ni (ε)) on the holed sphere Vi is by conformal diffeomorphisms. Then Liouville’s theorem implies that the diffeomorphisms are conformal diffeomorphisms of the round sphere S n , as n ≥ 3. In particular Ni (ε) is obtained from a spherical space form by deleting out several open disks. Remark 4.3. 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. This ends the proof of item (2) of the theorem, since we can obtain the manifold M by gluing together the essential pieces and the tubes which correspond to attaching 1-handles. Remark 4.4. The proof actually shows that M is conformally equivalent to a connected sum of spherical space forms and classical Schottky manifolds (see [32]). Remark 4.5. Let E be a connected component of M − C and E be its closure. Then, from ([10], Lemma 2.7) the boundary ∂E = E − E has at most two connected components and each such component is either a point, a (n − 1)-sphere or the union of two (n − 1)-spheres having a common tangency point.

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Remark 4.6. Kulkarni proved in [24] that closed orientable conformally flat manifolds which admit conformal embeddings as hypersurfaces in Rn+1 are conformally equivalent to either the round n-sphere or some Hopf manifold S 1 ×S n−1 , if both the immersion and the metric are analytic. The proof from above implies 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 . Furthermore one should notice that in [10, 32] the authors proved that closed conformally flat manifolds which admit conformal embeddings as hypersurfaces in some hyperbolic of Euclidean space are diffeomorphic and respectively conformally equivalent to an n-sphere with several (possibly unoriented) 1-handles. This shows many similarities with the first item of our theorem. 5. Proof of Theorem 1.3 5.1. Preliminaries concerning the positive λ. The aim of this section is to prove that components with positive λ cannot have non-trivial contributions to the topology of the manifold, other than cylinders over constant curvature space forms as in the case of local 1-QC manifolds. Lemma 5.1. If λ(p) > 0 for all p ∈ M − C then λ(p) ≥ λ0 > 0 for all p ∈ M − C. Proof. By continuity λ is bounded from below. If S is a curvature leaf of M having curvature λ then its image under the immersion is an embedded round sphere Sλ of curvature λ in Hn+1 κ . In particular, there exists a pair of points in S(λ) whose distance in Hn+1 , and hence also in the image of M is at least f (λ, κ) > 0, where limλ→0 f (λ, κ) = ∞. κ On the other hand, the preimage of these two points in M are also at distance at least f (λ, κ), on one side, and bounded from above by the diameter of M. It follows that λ is uniformly bounded away from zero. In general, for arbitrary 1-QC manifolds λ might have poles even when λ is positive. It suffices to consider a spherical cylinder with a disk 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: Lemma 5.2. If λ > 0 on M − C and λ is unbounded in a neighborhood of q ∈ C then q is an isolated isotropic point. Proof. The curvature leaves are round spheres Sλ whose diameter goes to 0. If their diameter is small enough they each such sphere Sλ bounds a disk Dλ on M. Now the subset Dλ − int(Dλ′ ), where Dλ′ ⊂ Dλ and λ′ > λ. is entirely foliated by curvature leaves, and hence it is diffeomorphic to a cylinder. The intersection of a nested sequence of disks is then a point. It follows that we can suppose from now on that there are two constants so that 0 < c ≤ λ(p) ≤ C < ∞, for all points p ∈ M.


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5.2. Neighborhoods of isotropic points. We only have to show that small neighborhoods of isotropic points are standard: Proposition 5.3. If ε is small enough then those Ni (ε) which have positive sectional curvature are diffeomorphic to spherical space forms with several disjoint open disks removed. Proof. The saturation lemma 4.7 shows that given ν > 0 there exists ǫ(ν) > 0 small enough such that |N(p) − H(p)| ≤ ν, for any p ∈ Ni (ε), meaning that Ni (ε) has pointwise pinched curvature. [ We wish to consider the manifold N i (ε) obtained topologically from Ni (ε) by capping off boundary spheres by disks and to show that it has also a pointwise pinched metric. Let S be a boundary component of Ni (ε) and F be a collar of S in Ni (ε) which is saturated by curvature leaves. In particular F is diffeomorphic to S n−1 × [0, 1]. As F is f and thus embedded into Hn+1 . We say that F simply connected it could be lifted to M κ has width w if the distance between the two boundary components is w. The argument used in the proof of lemma 4.3 shows that if we want that the result of capping off the collar F by some ball still be immersed into Hn+1 κ , then the measure of the cap is at least that of the hyperbolic ball Bλ lying in a n-plane, and thus uniformly bounded from below by the positive constant C1 (supp∈M λ, κ, n) from the proof of [ lemma 4.3. This will show that the volume of N i (ε) is uniformly bounded from below by C1 (λ, κ, n) > 0, if the capped collars are immersed in Hn+1 κ . We will show now that this can indeed be done. Consider then some leaf S in the boundary of the collar F and thus also in the boundary of some Ni (ε). Lemma 5.4. (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) whose boundary ∂H+ (S) is isometric to S and of radius q   κ r 1 + H(S)+κ 1 κ 1  q d = √ arctanh = √ log  κ H(S) + κ κ 2 κ 1− H(S)+κ

so that F ∪S H+ (S) embeds isometrically in Hn+1 κ . (2) If H(S) ≤ 0 then there is no hyperspherical cap tangent to F along S.

Proof. The image of F in Hn+1 is the envelope of a one-parameter family of hyperspheres, κ i.e. a canal hypersurface. This was already proved in [32, 12] 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 Hn+1 and ∇ be the Levi-Civita connection on Hn+1 κ κ . The Weingarten T (or shape) operator Lψ is defined as Lψ X = (∇X ψ) , where the superscript T means the tangent part. Moreover, g(Lψ X, Y ) = h(X, Y ), where X, Y are √ tangent to M and h is the second fundamental form of M. It follows that Lψ X = − H + κ · X for any vector field X tangent to the curvature leaf S.

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Lemma 5.5. Let S be a fixed round sphere which is the image of a curvature leaf. Then, either the focal locus of F is empty or else there is a point q ∈ Hn+1 such that the κ hypersphere H(S) of radius d and center q contains S and is tangent to F along S.

Proof. This seems to be widely known, except possibly for the explicit value of the radius when κ > 0 (see [12] for κ = 0). The first part follows from the case κ = 0, as that balls in Hn+1 are conformally equivalent to balls in the Euclidean space and curvatures leaves κ are preserved. Another, self-contained proof uses the particular form of the focal locus, which if nonempty it is a 1-dimensional manifold and hence a segment, as the focal points have multiplicity n − 1 (see [5]). An endpoint of the focal locus is at the same distance to any point of S, the corresponding boundary component of F since the derivative of the focal map ([5], section 1.e, Thm. 2.1 and proof of Thm. 3.1)) is trivial, so that S is contained κ in a hypersphere H(S). Notice that the focal map is only defined when H(S)+κ < 1, namely when H(S) > 0, corresponding to the fact that the extrinsic curvature has to be greater than κ. Still another proof of this statement could be found in [18] in a slightly different wording. In order to compute the radius of this hypersphere, we recall that for a unit speed geodesic γ parameterized by [0, l] issued from a point p and normal to the submanifold W of Hn+1 the Jacobi field Y (t) along γ is a W -Jacobi field if Y (0) is tangent to W at p, κ Y (t) is orthogonal to γ(t) ˙ for all t, and ∇γ(0) Y (0) + Lγ(0) Y (0) is orthogonal to W , where ˙ ˙ Lγ(0) is the Weingarten (or shape) operator of W . The point q on γ is called a focal point ˙ of W along γ if there exists a non-trivial W -Jacobi field along γ vanishing at q. If X1 , X2 , . . . , Xn is a set of parallel vector fields along γ which form along with γ˙ a n+1 basis of the ambient tangent space to Hn+1 along γ has the κ , then a Jacobi field on Hκ form (see [7]): n X √ √ (ai sinh κt + bi cosh κt)Xi (t), Y (t) = i=1

where ai , bi ∈ R. An immediate computation yields the fact that focal points of S along geodesics pointing in the direction of ψ exist if and only if H(S) > 0, in which case they are all at the same distance r 1 κ d = √ arctanh κ H +κ Moreover H(S) and F have the same normal vector field ψ at the points of S. Eventually S determines a spherical cap H+ (S) lying on H(S) such that its union with F along S is smooth.

Remark 5.1. Observe that when H(S) ≤ 0 the normal geodesics in the direction ±ψ diverge. Nevertheless there exists a totally umbilical embedded n-plane of constant negative intrinsic curvature H(S) which plays this time the role of the round sphere H(S). However, now the cap H+ (S) is unbounded. These are called equidistant planes (or hypersurfaces) and were first considered in [6]. In fact they are the locus of points at a


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given distance r from a hyperbolic hyperplane of Hn+1 κ , or a horosphere respectively. For instance, when H(S) = 0, H(S) is a horosphere centered at the point at infinity where all normal geodesics to S abut. Further, when H(S) < 0 we have an equidistant plane for the distance r = arcosh

√

1 |H(S)|

.

We restrict ourselves from now on to the case when H(S) > 0. We consider now [ Ni (ε) be the manifold obtained from Ni (ε) by adding to each boundary component S the corresponding cap H+ (S). This gives only a C 1 -metric on the union, but an arbitrarily small perturbation by a flat function along a small collar of the boundary will make it smooth. A neighborhood of S in Ni (ε) is a collar F , and by choosing ε small enough we can insure that the width w of the collar can be made arbitrarily small. Before to proceed we need now to introduce another metric invariant of a Riemannian manifold. Recall from [30] that the pointwise anisotropy of the metric is the quantity: I(p) = sup Kp (σ) − σ

R(p) , n(n − 1)

where Kp (σ) is the curvature of the plane σ at p and R(p) R is the scalar curvature at p. Furthermore the integral anisotropy of the manifold N is N I(p) d vol.

Lemma 5.6. Assume H(S) > 0. For n ≥ 3 there exists two constants C1 , C2 such that for any δ > 0 and for any for small enough ε we have [ (1) vol(N i (ε)) ≥ C1 > 0; 2 [ (2) diam (Ni (ε)) supp∈N\ |Kp (σ)| < C2 ; i (ε),σ [ (3) and the integral isotropy of N i (ε) is smaller than δ.

Proof. Proposition 5.4 shows that for small enough ε we can glue isometrically spherical [ caps along the boundary components to obtain a smooth Riemannian manifold N i (ε). + These caps are the complementary of H (S) within the sphere H(S), for each boundary component S of Ni (ε). The boundary components of Ni (ε) are either separating or non-separating. The separating ones are of two types: either the separated component is a manifold with non-trivial fundamental group or else it is simply connected. The boundary component is accordingly called non-trivial and respectively trivial. Each non-separating or separating non-trivial boundary component induces a nontrivial splitting of the fundamental group π1 (M). It follows that their total number is bounded from above by the maximal number r(π1 (M)) of factors in a free splitting of the fundamental group π1 (M). Further the number of trivial separating boundary components is uniformly bounded by C1 (supvol(M )λ,κ,n) according to remark 4.2. Moreover, each p∈M

[ cap lies on a hypersphere of radius d. It follows that the diameter of N i (ε) is uniformly

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bounded from above by C2 = C2 (vol(M), λ, H, κ, n) = r(π( M)) · diam(M) + 2πd ·

vol(M) , C1 (supp∈M λ, κ, n)

[ while the volume N i (ε) is at least C1 = C1 (λ, κ, n). Observe that d is uniformly bounded in terms of | inf p∈M H|. Now hyperspheres H(S) of radius d in Hn+1 have constant sectional curvature. As they κ √ κ are totally umbilical the principal curvatures are all equal to tanh(√κd) . It follows that the sectional curvature is equal to H(S). By choosing the smoothing collar of S small enough we obtain: sup |Kp (σ)| ≤ max(sup |H|, sup |N|) \ p∈N i (ε),σ

p∈M

p∈M

On the other hand the pointwise anisotropy I(p) is non-zero only on the the subset [ Ni (ε), as spherical caps Rhave constant curvature. Thus the integral anisotropy of N i (ε) is bounded by above by ν M |H|dvol. The claim of the proposition follows.

Now, the last ingredient in the proof of proposition 5.3 is the following deep result of Nikolaev (see [30]), which states precisely that given any constants C1 , C2 there exists some δ(C1 , C2 , n) > 0 such that any closed n-manifold N satisfying vol(N) ≥ C1 > 0, diam2 (N) supp∈N,σ |Kp (σ)| < C2 whose integral isotropy is smaller than δ is diffeomorphic [ to a space form. For small enough ε such that ν(ε)vol(M) < δ(C1 , C2 ) the manifold N i (ε) satisfies all conditions in Nikolaev’s criterion above. This ends the proof of item (3) of the theorem in the case where we have only essential pieces Ni (ε) with positive curvature. In this case choosing small enough collars we have only boundary components S with H(S) > 0. Now we can obtain the manifold M by gluing together the essential pieces and the tubes which correspond to attaching 1-handles. Remark 5.2. We don’t know whether essential pieces Ni (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 [15]) under bounded volume assumptions. This would be so if locally the dimension of C were at least 3, by F. Schur ([36]). However, the isotropic set C might contain points where the dimension is smaller than 3 also there might exist points in the same connected component which cannot be joined by a smooth curve within C, when the metric is not supposed real analytic. Remark 5.3. When λ takes some negative values on a connected component E of M − C it follows that λ should be negative on all of E, as hyperbolic closed manifolds cannot support a non-negative constant curvature metric. By the same reasons, when λ has zeroes it has to vanish identically. Thus isotropic points of negative curvature can only be approached by leaves with H(S) < 0 and the method considered above cannot provide an isometric capping off for the boundary components. For instance, if the negatively


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curved Gromov-Thurston manifolds (see [16]) admit a spine which can be realized by points of constant sectional curvature, then we could provide examples of 1-QC manifolds whose vertex manifolds are not space forms. These could not be realized as local 1QC manifolds. We also expect to have non-trivial contributions of negatively curved components E. Specifically, we can construct compactifications of warped product spaces by adjoining polyhedra and smoothing, but in general these have only a geodesic metric structure as Alexandrov spaces. 6. Some particular classes of 1-QC manifolds 6.1. Subprojective manifolds. The subprojective spaces are precisely the 1-QC manifolds whose distinguished vector field ξ is geodesic (see [12]). We keep calling them (local) subprojective spaces in our present setting of local 1-QC manifolds. Notice that the vector field ξ is (locally) conformally Killing if and only if it is Killing and iff ξ is geodesic. The structure of conformally flat closed manifolds whose universal covering have a conformal vector field was described by Goldman and Kamishima (see [14]). We formulate the following as an independent result since the proof is much simpler than the general case: Proposition 6.1. If M is subprojective each connected component E of M − C is a cylinder over some closed space form. Proof. Since the vector field ξ is geodesic we have for any horizontal vector field X that: dη(ξ, X) = g(∇ξ ξ, X) = 0 Therefore 0 = dη(ξ, X) = Xη(xi) + ξη(X) − η([ξ, X]) = −η([ξ, X])

so that [ξ, D] ⊂ D. It is known (see [3], Prop. 1.2.21) that this a necessary and sufficient condition for the one parameter flow φξ,t associated to the vector field ξ to send diffeomorphically leaves onto leaves. Eventually recall from above that, whenever H is not constant on E almost all leaves are compact, so that all of them should be compact. When H is constant the component E is still diffeomorphic to L × (0, 1), where L is an arbitrary leaf. However, in this case we also know (see [2, 12]) that the leaves are totally geodesic, i.e. α = 0. Furthermore η is closed and moreover by ([12], equations (3.8), (3.12)) we have N = −ξ(α) − α2 = 0

When the leaves L approach a point p ∈ C the functions H and N approach the constant curvature value of the isotropic point p. If H 6= 0 this would lead to a contradiction, so that necessarily H = N = 0. But then the points in E would be also isotropic, contradicting our assumption that E ⊂ M − C.

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Remark 6.1. It follows from ([12], Prop.7) that E must be isometric to a warped product L ×f (0, 1), with metric gρ for some function ρ : (0, 1) → R∗ , namely gρ (X, Y ) = ρ2 g0 (X − η(X)ξ, Y − η(Y )ξ) + η(X)η(Y )

where g0 is a constant curvature metric on L, whose sign is the sign of λ. Here ρ is determined ξ(ρ) ξ(H) = ρ 2(N − H) 6.2. The space of 1-QC metrics. The connected components of the interior int(C) of the set of isotropic points have will be called islands in the sequel. Of course, according to Schur’s isotropy theorem islands have constant sectional curvature and thus a well-defined sign. This also holds for the closure of each connected component, namely for int(C). The meaning of theorem 1.3 is that when λ > 0 and we have only positive islands then the manifold has a standard topology. Further also their geometry is well-understood around the islands. In fact, we proved above that globally 1-QC spaces with positive H(S) are channel hypersurfaces (i.e. envelopes of one-parameter families of spheres) in Hn+1 κ . When the manifold is local 1-QC, then each channel hypersurface abuts to a limit f of the closure of a positive leaf L. Since the inverse image into universal covering M island must be diffeomorphic to a round sphere Σ minus several round balls it follows that both H(S) and this sphere Σ should be tangent to each other along the limit leaf f is L. As the limit leaf is a round sphere then H(S) coincides with Σ. It follows that M a closed channel hypersurfaces, namely the envelope of a family of spheres parameterized by a graph, for which vertices correspond to the islands. Now, for a general situation of a local 1-QC manifold the situation is similar, as limit leaves separate isotropic points (which are closures of islands) and globally 1-QC components which are topologically cylinders. Therefore the set of all 1-QC metrics would be given by smooth functions H, N defined on these graphs. In the case when a global 1-QC component is subprojective we have a warped product [0, 1] × L, where L is of constant curvature k0 and the warping function ϕ : [0, 1] → (0, ∞). Then the sectional curvature functions of the slices is given by: H(t) =

k0 − ϕ′ (t)2 ϕ′′ (t) , N(t) = − ϕ(t)2 ϕ(t)

In this case N is determined by H. For general global 1-QC metrics it seems that H and N should rather satisfy systems of inequalities, and N is not necessarily constant on the slices, but only H and λ. 6.3. Another positivity condition. Proposition 6.2. Assume that (n − 2)H + 2N ≥ 0 for all points of the compact 1-QC n-manifold M n . Then M n is conformally equivalent to a Kleinian quotient Ω/G where G is a discrete subgroup of O(n + 1, 1), the Mobius group acting on S n and Ω is its discontinuity domain. Moreover πj (M) = 0, for 2 ≤ j ≤ n/2.


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Proof. In [35] Schoen and Yau proved that for locally conformally flat compact M the scalar curvature of a suitable metric in the conformal class has constant sign. This sign is well-defined and it is given by the sign of the Yamabe invariant of the manifold. If this sign is negative then the fundamental group π1 (M) is non-amenable. If the sign of the scalar curvature is nonnegative ([35], Thm 4.5) then M is conformally equivalent to a Kleinian quotient Ω/G where G is a discrete subgroup of O(n + 1, 1), and Ω is its discontinuity domain. Moreover, if the scalar curvature is strictly positive then πj (M) = 0, for 2 ≤ j ≤ n/2, (see [35], Thm 4.6). By passing to a double cover it suffices to consider the case when M is orientable. The sign of the Yamabe invariant is the sign of the first eigenvalue of the conformal Laplacian n−2 Lu = ∆u + R·u 4(n − 1) where ∆u = −div grad u is the Laplace-Beltrami operator and R the scalar curvature. Since ∆ has positive eigenvalues for compact M it follows that when R ≥ 0 the first eigenvalue of L is also strictly positive. Indeed, if φ is a (non-zero) eigenfunction of L then Z Z Z n−2 2 R · φ2 > 0 φLφ = |grad φ| + 4(n − 1) M M M 7. General 1-QC manifolds 7.1. Equivariant isometric immersions. The immersability result from proposition 2.3 can be easily improved as follows: Proposition 7.1. If M n is connected 1-QC manifold then there exist an equivariant f → Hn+1 , and a group hoisometric immersion, namely an isometric immersion f : M κ momorphism ρ : π1 (M) → SO(n + 1, 1) such that f, γ ∈ π1 (M) f (γ · x) = ρ(γ) · f (x), x ∈ M where the action on the right side is the action by isometries of SO(n + 1, 1) on Hn+1 κ while the left side action is by deck transformations on the universal covering.

f is invariant by the action of the deck group, Proof. The second fundamental form h on M 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 xi in general position in Hn+1 κ , which belong to a small open neighborhood of x into their respective images γ · xi . 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 isometric immersion is now discrete. The first and simplest case is:

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Proposition 7.2. Suppose that n ≥ 4. If κ is such that H + κ > 0 on M there exists f → Hn+1 , up to an isometry (possibly reversing the an unique isometric immersion f : M κ n+1 orientation) of Hκ . 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 and make it coincide with the one from the proof of proposition 2.3. Now an isometric immersion is uniquely determined by its second fundamental form. Remark 7.1. The same argument shows that there does not exist any isometric immersion around a point where H + κ ≤ 0. Proposition 7.3. Suppose that n = 3 and κ is such that H + κ > 0 on M. f → Hn+1 (1) If N + κ 6= 0 then there exists an unique isometric immersion f : M κ , up to n+1 an isometry of Hκ . (2) Assume that N + κ = 0. Then the set of such equivariant isometric immersions of a component E of M − C is in bijection with the set of paths in the space of complex structures over the surface L corresponding to a curvature leaf. Proof. If {X1 , X2 , X3 = ξ} is a local orthonormal basis and hij are components of the second fundamental form of the immersion then Gauss equations read: h11 h22 − h212 = µ, h11 h33 − h213 = ν, h22 h33 − h223 = ν h23 h11 = h12 h13 , h13 h11 = h12 h23 , h12 h33 = h13 h23 where µ = H + κ, ν = N + κ. The homogeneous equations above imply that either h11 = h22 or else ν(h11 + h22 ) = µh33 . The first alternative leads to the immersion arising from the form h in proposition 2.3, as happened for n = 4. The second one coupled with the non-homogeneous equations leads to ν = 0. In this case we obtain the solution consisting of hij with h33 = h13 = h23 = 0 while the other components are satisfying: h11 h22 − h212 = µ

It follows that the restriction of h to each lift π −1 (L) of a curvature leaf L ⊂ E (of constant f is the second fundamental form of an immersion fλ0 : π −1 (L) → Hn . curvature H0 ) in M κ f → Hn+1 , with respect to the foliation This immersion fλ0 is a slice of the immersion f : M κ of Hn+1 by hyperbolic hyperplanes Hnκ . κ In ([26], Prop. 2.5) one identified the 2-tensors verifying the Gauss and Codazzi equations for a constant curvature metric on a surface with the complex structures on the surface. Therefore, the complex structures of the slices of E give us a path of complex structures on the slice surface. Recall that the sectional curvature of the slice surface is the not necessarily constant function λ on E, so that we need to rescaled it in every slice in order to fit exactly into the framework of ([26], Prop 2.5).


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f → Hn+1 is covered by an equivariant immersion into Observe that the immersion f : M κ n+1 ˆ f the (unit) tangent bundle f : M → T Hκ , defined by fˆ(p) = (f (p), nf (p) ) ∈ T Hn+1 κ

f) at the point f (p) associated where nf (p) denotes the unit (positive) normal vector at f (M to the local sheet defined around p. Let us further consider the hyperbolic Gauss map G : T Hn+1 → S n which sends the pair κ n+1 (p, v), where p ∈ Hn+1 to the point at infinity to which the geodesic issued κ , v ∈ Tp Hκ from p in direction v will reach the boundary. Here S n is identified with the boundary at infinity ∂Hn+1 of the hyperbolic space. κ Proposition 7.4. Suppose that H + κ > 0, N + κ > 0. The composition F = G ◦ fˆ : f → S n is an immersion. Moreover the flat conformal structure defined by the 1-QC M metric on M is the one given by the developing map F and the holonomy homomorphism ρ. f) is locally convex, when H + κ > 0, Proof. The first part follows from the fact that f (M N + κ > 0. Moreover, by the same argument as above (for n ≥ 2) the immersion F is equivariant with respect to the same homomorphism ρ : π1 (M) → SO(n + 1, 1) as above, but this time interpreting SO(n+1, 1) as the group of Möbius (conformal) transformations of the sphere. It follows that the conformally flat structure on M is actually given by the developing map F and the holonomy homomorphism ρ. Proposition 7.5. If n ≥ 3, then there exists a canonical hyperbolic metric on M ×[0, ∞). f→ Proof. This is already standard, but here is a simple proof. Using the immersion f : M n+1 Hκ we can construct under the conditions of proposition 7.4 an equivariant immersion f × [0, ∞) → Hn+1 by φ:M κ φ(p, t) = exp(tnf (p) ) f) is locally convex. This is well-defined for all t ∈ R, and it is an immersion when f (M ∗ induced by this immersion as In particular we have a constant curvature metric φ gHn+1 κ n+1 n+1 pull-back of the usual metric gHκ on Hκ . Since F is equivariant the metric descends to M × [0, ∞) and it becomes hyperbolic after rescaling. Remark 7.2. In order to apply proposition 7.4 we have to choose κ large enough. However a different κ will lead to the same hyperbolic metric. Remark 7.3. When n = 2 the situation is different, and it was described by Labourie in [26]. The space I(k0 ) of codimension one isometric equivariant isometries of a surface of a surface S of genus at least 2 into the hyperbolic space H31 , such that the pull-back metric is of constant curvature k0 ∈ (−1, 0) – up to left composition by isometries of H31 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(k0 ) into the Teichm¨ uller space T (S) which associates to an immersion the conformal class ι0 of the induced constant curvature

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metric. The fiber I(k0 , ι0 ) of this map can also be mapped homeomorphically onto the Teichm¨ uller 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(k0 ) 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 parameterizations 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 fk : S → H31 , whose associated CP 1 structure is the one with which we started. Further, the immersion φk associated to fk provides a hyperbolic structure M(k) on S × [0, ∞), called a geometrically finite end (see [26]). In [26, 27] one proved that for each constant k ∈ [k0 , 0) there exists a unique incompressible embedded surface Sk ⊂ S × [0, ∞) of constant curvature k homeomorphic to S and the family Sk foliate the geometrically finite end M(k0 ). This proves that actually M(k1 ) ⊂ M(k2 ) if k1 > k2 . 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 H31 is a pleated surface along a geodesic lamination, according to Thurston (see [26] for more details). Higher dimensional versions of Labourie’s results concerning moduli of hyperbolic flat conformal were obtained by Smith in [37]. References [1] V.Boju and L.Funar, Espaces ` a courbure Stanilov quasi-constante, Serdica Bulg. Math. Publ. 9(1983), 307–308. [2] V.Boju and M.Popescu, Espaces ` a courbure quasi-constante, J. Diff. Geometry 13(1978), 373–383. [3] A.Candel and L.Conlon, Foliations, vol 1. Graduate Studies in Mathematics, 23. American Mathematical Society, Providence, RI, 2000. xiv+402 pp. [4] E. Cartan, La déformation des hypersurfaces dans l’espace conforme réel ` a n ≥ 5 dimensions, Bull. Soc. Math. France 45 (1917), 57–121. [5] T.E.Cecil and P.J.Ryan, Focals sets of submanifolds, Pacific J. Math. 78(1978), 27–39. [6] T.E.Cecil and P.J.Ryan, Distance functions and umbilic submanifolds of hyperbolic space, Nagoya Math. J. 74 (1979), 67–75. [7] J.Cheeger and D.G. Ebin, Comparison theorems in Riemannian geometry, Revised reprint of the 1975 original. AMS Chelsea Publishing, Providence, RI, 2008. x+168 pp. [8] B.-Y.Chen and K.Yano, Special conformally flat spaces and canal surfaces, Tohoku J. Math. 25(1973), 177–184. [9] S.S.Chern and J.Simons, Characteristic forms and geometric invariants, Ann. Math. 99 (1974), 48–69. [10] M. Do Carmo, M. Dajczer and F. Mercuri, Compact conformally flat hypersurfaces, Trans. Amer. Math. Soc. 288(1985), 189–203. [11] R.Edwards, K.Millett and D. Sullivan, Foliations with all leaves compact, Topology 16 (1977), no. 1, 13–32. [12] G.Ganchev and V.Mihova, Riemannian manifolds of quasi-constant sectional curvatures, J. reine angew. Math. 522(2000), 119–141. [13] W. Goldman, Conformally flat manifolds with nilpoetent holonomy and the uniformization problem for 3-manifolds, Trans. Amer. Math. Soc. 278 (1983), 573–583.


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