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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)

53C20 Global Riemannian geometry, including pinching
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