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A canonical metric for Möbius structures and its applications. (English) Zbl 0813.53022
A Möbius structure on a manifold $$M^ n$$ is by definition a geometric structure modeled with $$S^ n$$ on which the full group $${\mathcal M}(n)$$ of conformal transformations, that is, the Möbius group, acts freely. Examples are given by conformally flat Riemannian manifolds, space-forms or $$CP^ 1$$-structures. Clearly, $$S^ n$$ has a canonical Möbius structure, which one denotes by $$\sigma_ 0$$. If $$\sigma$$ is such a structure on any simply connected manifold $$M^ n$$, then it admits a development map $$\text{dev}: M^ n\to S^ n$$ satisfying $$\sigma= \text{dev } ^*\sigma_ 0$$. Now, let $$\widetilde {M}^ n$$ be the universal covering of $$M^ n$$. $$M^ n$$ is said to be developable if $$\text{dev}: \widetilde {M}^ n\to S^ n$$ factors through the covering projection $$\widetilde {M}^ n\to M^ n$$. For this case one attaches an ideal boundary $$\partial_ 0 (M^ n)$$, and if it is empty, or a single point, $$M^ n$$ is called elliptic or parabolic, respectively, and hyperbolic otherwise. For a hyperbolic manifold structure $$M^ n$$ we have the theorem: It admits a complete Riemannian metric with Lipschitz derivatives. It also admits a canonical stratification by totally geodesic hyperbolic Riemannian manifolds of dimension $$k$$, $$1\leq k\leq n$$. The paper is now devoted to construct a metric on the general (and the hyperbolic) Möbius structure.
Close to the end, in Section 8, the notion of $$H$$-hull is introduced to a developable Möbius manifold structure $$M$$. Its boundary is actually the pleated image $$P(M)$$ of $$M$$, for which one has the following theorem: there is a natural map $$\sigma: M\to P(M)$$ which is an isometry on each stratum. Moreover, there is an 1-1 correspondence between simple connected hyperbolic Möbius manifolds and simply connected pleated $$n$$- sets.
Reviewer: T.Okubo (Victoria)

##### MSC:
 53C20 Global Riemannian geometry, including pinching
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