KUIPER’S THEOREM ON CONFORMALLY FLAT MANIFOLDS

RALPH HOWARD

DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA COLUMBIA, S.C. 29208, USA

                             HOWARD@MATH.SC.EDU

1. Introduction

These are notes to that show how to modify the proof given by Kuiper [2] that a compact simply connected conformally flat manifold is conformally diffeomorphic to a sphere under less restrictive smoothness conditions (Kuiper works with metrics of class C^{3}). The proof mostly follows the original proof of Kuiper other than we use a covering space argument rather than his monodromy argument and we restrict ourselves to Riemannian metrics while Kuiper works with conformally flat semi-Riemannian manifolds. What allows us to extend Kuiper’s proof is a theorem of Gehring [1] which shows that the theorem of Louisville on conformal maps between Euclidean space of dimension three or more holds for C^{1} maps.

2. Regularity of Conformal Maps.

By a C^{1} conformally flat manifold we mean a Riemannian manifold (M, g) so that M is of class C^{1}, the metric is of class C^{0} and every point has a C^{1} coordinate system x^{1}, …, x^{n} so that in this coordinate path the metric has the form g = λ^{2}((dx^{1})^{2} + ⋯ + (dx^{n})^{2}). Or what is the same thing that M has a cover by open sets {U_{α}} which are the domain of C^{1} diffeomorphisms φ_{α}: U_{α} → φ_{α}[U_{α}] ⊂ R^{n} onto open sets so that the transition functions Φ_{α,β} := φ_{α} ∘ φ_{β}^{−1}|_{φ_{β}[U_{α}∩U_{β}]} : φ_{β}[U_{α} ∩ U_{β}] → φ_{α}[U_{α} ∩ U_{β}] are of class C^{1} and if g_{0} is the flat metric on R^{n} then φ_{α}^{∗}g_{0} = λ_{α}^{2}g for some positive continuous function λ_{α} defined on U_{α}.

As the sphere S^{n} with its standard metric is locally conformally flat we could just as well as taken the maps φ_{α} to have values in S^{n} rather than R^{n} and in what follows we will often do this.

We now recall the definition of a Möbius transformation of the sphere S^{n}. Let R_{1}^{n+2} be R^{n+2} with the Lorentzian inner product g_{1}^{n} = (dx^{1})^{2} + ⋯ + (dx^{n+1})^{2} − (dx^{n+2})^{2} and let O^{+}(n + 2, 1) be the group of linear maps that preserve both the inner product and the direction of the “time axis” x^{n+2}. Embed S^{n} in R_{1}^{n+2} as S^{n} := {(x, 1) ∈ R^{n+1} × R : ∥x∥ = 1} where ∥x∥^{2} = (x^{1})^{2} + ⋯ + (x^{n+1})^{2} is the usual Euclidean norm on R^{n+1}. Then the induced metric on S^{n} is the usual round metric. Each a ∈ O^{+}(n + 1, 1) a map Ψ_{a} on S^{n} as follows. Let a(x, 1) = (a_{1}x, a_{2}(x)). Then as a preserves the time orientation the real number a_{2}(x) is positive. Set

Date: July 1996.

Ψ_{a}(x, 1) = (\frac{a_{1}(x)}{a_{2}(x)}, 1) .

Then Ψ_{a}Ψ_{b} = Ψ_{ab} and each of the maps Ψ_{a} is conformal on S^{n}. (To see this note that x ↦ (a_{1}(x), a_{2}(x)) is an isometry of S^{n} with a[S^{n}] as a is an isometry of the ambient space R_{1}^{n+2} and (a_{1}(x), a_{2}(x)) ↦ a_{2}(x)^{−1}(a_{1}(x), a_{2}(x) = Ψ_{a}(x, 1) is then a conformal map.) The maps Ψ_{a} with a ∈ O^{+}(n + 2, 1) are Möbius transformations. The collection of all Möbius transformations is then a group of conformal transformations acting on S^{n} called the Möbius group. The proof of Kuiper’s theorem is based on

Theorem 2.1 (Louisville-Gehring). Assume n ≥ 3 and that U ⊂ S^{n} is open connected set. Then any C^{1} conformal map φ: U → S^{n} is the restriction of a Möbous transformation.

Proof. This is Theorem 16 on page 389 of Gehring’s paper [1]. Louisville had proven the result under the assumption the map is of class C^{4}. Gehring reduces the more general result to this case by proving a regularity theorem which implies that a C^{1} conformal map is real analytic. In fact the result of Gehring is even more general than the C^{1} result as he has a definition of what it means for a continuous map to be conformal and then proves his regularity theorem for maps that conformal in this sense except on a set of finite n − 1 dimensional measure. Also Gehring works in R^{n} instead of S^{n}, but the S^{n} results follows easily by use of stereographic projection.

Here we give a different proof that, while not quite rigorous, uses the machinery of Riemannian geometry rather than that of quasiconformal maps. We first recall some facts about the conformal Laplacian. Let g and \overline{g} be two conformal metrics on manifold M, say \overline{g} = λ^{2}g. Let S be the scalar curvature and Δ the Laplacian of g and \overline{S} and \overline{Δ} the scalar curvature of \overline{g}. Then for any smooth function u on M

(\overline{Δ} − \frac{(n − 2)\overline{S}}{4n − 4} ) (λ^{\frac{2−n}{2}} u) = λ^{− \frac{n+2}{2}} (Δ − \frac{(n − 2)S}{4n − 4} ) u.

As a special case let M be a connected open set in R^{n} and let φ: U → R^{n} be conformal, so that for some positive function λ: U → R there holds φ^{∗}g = λ^{2}g. For the time being we don’t worry about the smoothness of φ and just assume that it has all the derivatives we need. Let \overline{g} := φ^{∗}g = λ^{2}g. Then \overline{g} is the pullback of the flat metric and so it is also flat. Therefore both g and \overline{g} have zero scalar curvature. Therefore in our case the last equation becomes

\overline{Δ}(λ^{\frac{2−n}{2}} u) = λ^{− \frac{n+2}{2}} Δu. (2.1)

Now let u := λ^{\frac{n−2}{2}} so that λ^{\frac{2−n}{2}} u ≡ 1. Then equation (2.1) and \overline{Δ}1 = 0 yields Δλ^{\frac{n−2}{2}} = 0 so that by Weyl’s lemma λ^{\frac{n−2}{2}} and therefore also λ is real analytic. Write φ = (φ^{1}, …, φ^{n}) so that φ^{i} = φ^{∗}x^{i}. Then as the coordinate functions x^{i} are harmonic with respect to the flat metric the functions φ^{i} are harmonic with respect to the metric \overline{g}. Therefore \overline{Δ}φ^{i} = 0. So if we let u = λ^{\frac{n−2}{2}} φ^{i} in (2.1) we find Δ(λ^{\frac{n−2}{2}} φ^{i}) = 0 so that φ^{i} is also real analytic. Whence φ is real analytic.

To make this all rigorous in the case φ is only C^{1} it is enough to show that all all the statements can be interpreted in the weak sense and that they still hold in this sense when φ is only C^{1}. While I have not done this my deep faith in elliptic technology makes me believe it works.

Corollary 2.2. Any C^{1} conformally flat manifold (M, g) has a natural real analytic structure and there is a C^{∞} metric in the conformal class of g.

Proof. Let {U_{α}} an open cover of M so that there are C^{1} conformal maps φ_{α}: U_{α} → φ_{α}[U_{α}] ⊂ S^{n} as in the definition of a C^{1} conformal manifold above and let Φ_{α,β} := φ_{α} ∘ φ_{β}^{−1}|_{φ_{β}[U_{α}∩U_{β}]} : φ_{β}[U_{α} ∩ U_{β}] → φ_{α}[U_{α} ∩ U_{β}] be the corresponding transition functions. Then by the theorem of Louisville-Gehring the restriction of Φ_{α,β} to any connected component of φ_{β}[U_{α}∩U_{β}] is the restriction of a Möbius transformation and therefore Φ_{α,β} is real analytic. This gives a real analytic atlas on M.

Let g_{0} be the standard metric on S^{n}. The metric g_{α} := φ_{α}^{∗}g_{0} on U_{α} is real analytic with respect to the real analytic structure we have defined on M and is also conformal to the metric g on U_{α} on U_{α}. These metrics can be pieced together by use of a partition of unity to give a smooth metric in the conformal class of g.

3. Kuiper’s Theorem

Theorem 3.1 (Kuiper [2]). Let (M^{n}, g) be a simply connected conformally flat manifold of class C^{1}. Then there is a conformal immersion f: M → S^{n}. If M is compact then this map is a conformal diffeomorphism of M with S^{n}.

Proof. Let U = {U_{α}} be an open cover of M by connected open sets so that for each α there is a C^{1} injective conformal map φ_{α}: U_{α} → S^{n}. We now claim that we can find anther open cover V and for each V ∈ V an injective conformal map ψ_{V} : V → S^{n} so that V is countable, each V ∈ V and each intersection V_{1} ∩ V_{2} with V_{1}, V_{2} ∈ V is connected. To see this we use that M is paracompact so that we can express M = ∪_{i=1}^{∞}K_{i} as a countable union of compact sets K_{i} and so that this union is locally finite. Now choose a complete Riemannian metric h on M (which need not be related to the conformal structure). For each i let δ_{i,1} be the Lebesgue number of the cover on K_{i}. That is if r ≤ δ_{i,1} and x ∈ K_{i} then the open ball B(x, r) is a subset of some member U_{α} of U. Let δ_{i,2} be the convexity radius of K_{i} in M, that is if x ∈ K_{i} and r ≤ δ_{i,2} then the ball B(x, r) is convex in the sense that any two points of B(x, r) can be joined by a unique minimizing segment and this segment is a subset of B(x, r). Let δ_{i} := min(δ_{i,1}, δ_{i,2}). Then {B(x, δ_{i}) : x ∈ K_{i}} is an open cover of K_{i} and therefore it has a finite sub-cover V_{i}. For each V ∈ V_{i} by construction there is a U_{α} ∈ U so that V ⊆ U_{α}. Set ψ_{V} = φ_{α}|_{V}. Finally set V := ∪_{i=1}^{∞}V_{i}. This is a countable union of finite sets and thus countable and the connectivity of the intersection V_{1} ∩ V_{2} follows from the convexity of V_{1} and V_{2}.

Let V_{1}, V_{2} ∈ V and V_{1} ∩ V_{2} ≠ ∅ then by the theorem of Louisville-Gehring the transition function Ψ_{V_{1},V_{2}} := ψ_{V_{2}}∘ψ_{V_{1}} |_{ψ_{V_{1}}[V_{1}∩V_{2}]} : ψ_{V_{1}}[V_{1}∩V_{2}] → ψ_{V_{2}}[V_{1}∩ V_{2}] is the restriction of a Möbius transformation. This implies there is a unique Möbius transformation a so that ψ_{V_{1}} and a ∘ ψ_{V_{2}} agree on the set V_{1} ∩ V_{2}. We then say that a ∘ ψ_{V_{2}} is the analytic continuation of ψ_{V_{1}} into V_{2}. More generally given a chain of open V_{0}, V_{1}, …, V_{k} ∈ V (by a chain we mean that V_{i} ∩ V_{i+1} ≠ ∅ for i = 0, …, k − 1) we can repeat this and get a unique analytic continuation of ψ_{V_{0}} to the set V_{k} along the given chain. We now want to show that when M is simply connected this analytic continuation is independent of the chain connecting V_{0} and V_{k}.

Toward this end we fix a V_{0} ∈ V to use as a starting point for our construction. For any V ∈ V let C_{V} be the collection of all analytic continuations of ψ_{V_{0}} to V any chain V_{0}, …, V_{k} = V connecting V_{0} and V. Define a submanifold G ⊂ M × S^{n} by

G := ⋃_{V ∈V}{(x, ψ(x)) : x ∈ V, ψ ∈ C_{V} },

where the manifold structure is defined so that the projection onto M is a local diffeomorphism. If V_{0}, …, V_{k} = V is a chain and ψ_{V_{0}} =: φ_{0}, φ_{1}, …, φ_{k} are the maps obtained by analytically continuing ψ_{V_{0}} along the chain. Then ∪_{i=0}^{k}{(x, φ_{i}(x)) : x ∈ V_{i}} is a connected subset of G. But from the definition G this means that every point of G is in the same connected component as (x, ψ_{0}(x)) for x ∈ V_{0} thus G is connected. Let π: G → M be the restriction of projection onto the first factor. Then for each V ∈ V the set π^{−1}[V] is the disjoint^{1} union of the sets {(x, ψ(x)) : x ∈ V } where ψ varies over C_{V} and the restriction of π to any of one of the sets {(x, ψ(x)) : x ∈ V } is a diffeomorphism with V. Thus each of the sets V is evenly covered by the map π: G → M. Thus π: G → M is a covering map. As we are assuming that M is simply connected and we have shown G is connected this means that π: G → M is a diffeomorphism. Therefore G is the graph of a function f: M → S^{n} which is easily seen to be a conformal immersion.

Finally assume that M is compact. As the map f: M → S^{n} is a conformal immersion the image of f is an open subset of S^{n}. But M compact implies it is also closed. As S^{n} is connected the map f is surjective. For any point x ∈ M there is a neighborhood U of x so that f is injective on U. By the compactness of M we can then cover M by open sets U_{1}, …, U_{k} so that f is injective on each U_{i}. This implies any point y ∈ S^{n} has at most k preimages. So let f^{−1}[y] := {x_{1}, …, x_{l}}. Let N_{i} be a open neighborhood of x_{i} so that N_{1}, …, N_{l} are pairwise disjoint and so that f is injective on each N_{i}. Then U := ∩_{i=1}^{1}f[N_{i}] is an open neighborhood of y and f^{−1}[U] := ∪_{i=1}^{l}(f^{−1}[U]∩N_{i}) and f|_{f^{−1}[U]∩N_{i}}: f^{−1}[U]∩N_{i} → U is a diffeomorphism for all i = 1, …, l. Therefore f is a covering map and as S^{n} is simply connected this means it is a diffeomorphism.

1Strictly speaking to insure that this union is disjoint we should not use the submanifold topology on G, but rather the topology in inherits as subset set of the sheaf of germs of smooth S^{n} valued functions.

References

1. F. W. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353–391.

2. N. H. Kuiper, On conformally flat spaces in the large, Ann. Math. 50 (1949), 916–924.