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Singular metrics of nonpositive curvature on branched covers of Riemannian manifolds. (English) Zbl 0804.53056

This paper deals with the \(\text{CAT}(\chi)\)-properties of geodesic spaces, a generalization of the notion of “curvature \(\leq\chi\)”. Basic material is originally due to M. Gromov and has been elaborated – among others – by H. Haefliger and W. Ballmann (notoriously misspelled as “Ballman”), see E. Ghys and P. de la Harpe (eds.) [Sur les groupes hyperboliques d’après Mikhael Gromov, Prog. Math. 83, Birkhäuser (1990; Zbl 0731.20025)]. In particular, the paper aims at branched coverings \(\pi: \widetilde M\to M\) and the pullback of a metric with bounded curvature on \(M\). Along the branch locus, this pullback metric on \(\widetilde M\) is necessarily singular if \(M\) is smooth. Chapter I: Orbifolds of nonpositive curvature deals with the CAT(0)- property for which five crucial conditions (A), (B), (C), (D), (E) are developed. The authors conjecture that these five properties are satisfied – up to orthogonal Cartesian products – only in four irreducible cases. In terms of the group \(G\) of deck transformation around the branch locus, these cases are: 1. the trivial group acting on \(\mathbb{R}^ 1\); 2. the cyclic group acting on \(\mathbb{C}^ 1\); 3. the complex reflection groups \(G(p,q,r)\) acting on \(\mathbb{C}^ 2\), where \({1\over p}+ {1\over q}+ {1\over r}>1\); 4. on orientation-preserving subgroup of a real reflection group acting on \(\mathbb{R}^ n\). These group actions can be regarded also on the unit sphere. Consequently, Chapter II: Real and complex reflection groups focuses on the CAT(1)-property. In the case of the \(G(p,q,r)\) the branch locus turns out to consist of three Hopf circles whose images under the Hopf mapping \(H: S^ 3\to S^ 2\) lie on one great circle with mutual distances determined by \(p\), \(q\), \(r\). This theorem 9.1 is the main result of the paper. Its proof is fairly long.

MSC:

53C20 Global Riemannian geometry, including pinching
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