## Conformally flat manifolds whose development maps are not surjective. I.(English)Zbl 0608.53036

A Riemannian manifold M is conformally flat if for each point m of M there exist open sets $$U\subseteq M$$ and $$O\subseteq R^ n$$ and a conformal diffeomorpism $$x: O\to U$$ with $$m\in U=x(O)$$. For conformally flat manifolds one has the following uniformization theorem that follows from work of N. Kuiper and R. Kulkarni: Theorem. Let M be a smooth manifold of dimension $$n\geq 3$$, and regard M as a quotient space $$X/\pi$$ where X is the universal Riemannian cover of M and $$\pi \subseteq Isom(X)$$ is the fundamental group of M. If M is conformally flat, then there exists a conformal immersion $$d: X\to S^ n$$ and a homomorphism $$\phi: \pi\to Conf(S^ n)$$ such that $$\phi (\gamma)\circ d=d\circ \gamma$$ for all elements $$\gamma$$ of $$\pi$$. The pair $$(d,\phi)$$ are unique up to conjugation by an element of $$Conf(S^ n)$$, the group of conformal transformations of the sphere $$S^ n.$$
The group $$\phi(\pi) \subseteq Conf(S^ n)$$ is called the holonomy group of M. The group $$Conf(S^ n)$$ may be identified with $$SO(1,n+1)$$, the connected group of isometries of the $$n+1$$ dimensional hyperbolic space $$H^{n+1}$$. In fact, if one regards $$H^{n+1}$$ as the open unit ball in $$R^{n+1}$$ together with a suitable metric (conformally equivalent to the Euclidean metric), then the isometries of $$H^{n+1}$$ extend uniquely to conformal maps of $$S^ n$$ and vice versa, where $$S^ n$$ is regarded both as the topological boundary of $$H^{n+1}$$ in $$R^{n+1}$$ and as the geometric boundary $$H^{n+1}(\infty)$$ consisting of equivalence classes of asymptotic geodesics of $$H^{n+1}$$. Now define $$\partial (dX)$$ to be $$\overline{dX}-dX$$, where dX denotes the image of X in $$S^ n$$ under the conformal immersion d above. By construction the set $$\partial (dX)$$ is closed and invariant under the holonomy group $$\Gamma =\phi (\pi)$$. If $$\partial (dX)$$ is a single point, then M is flat and $$\phi$$ : $$\Gamma \to \pi$$ is an isomorphism by a result of Fried. Otherwise $$\partial (dX)$$ lies in the hyperbolic limit set $$L(\Gamma)\subset H^{n+1}(\infty)$$ and equals $$L(\Gamma)$$ if $$\partial (dX)$$ has at least 3 points.
Using facts from hyperbolic geometry about $$L(\Gamma)$$ the author obtains the following results. Theorem A. Let M be a 3-dimensional closed conformally flat manifold whose development map $$d: X\to S^ n$$ is not surjective. Then either the holonomy group is discrete or M is a locally homogeneous Riemannian manifold of nonpositive sectional curvature. In the latter case some finite covering of M is diffeomorphic to a product $$S^ 1\times N_ g$$, where $$N_ g$$ denotes a compact surface of genus $$g\geq 2$$. Theorem B. Let M be a closed conformally flat manifold of dimension $$n\geq 3$$. If the development map $$d: X\to S^ n$$ is not surjective, then d is a covering map. Theorem C. Let M be a closed conformally flat manifold of dimension $$n\geq 3$$. If the holonomy group contains a solvable subgroup of finite index, then M is a spherical space form, a Euclidean space form or a non-negatively curved manifold finitely covered by a Hopf manifold $$S^ 1\times S^{n-1}$$. Theorem C is related to work of W. Goldman.
Reviewer: P.Eberlein

### MSC:

 53C20 Global Riemannian geometry, including pinching 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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### References:

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