Ptolemaic spaces and CAT(0). (English) Zbl 1171.53024
The authors investigated the relation between Ptolemaic spaces and CAT(0) spaces. If CAT(0) spaces are Ptolemaic, the converse is not true. Still, a complete Riemannian manifold is Ptolemaic if and only if it is CAT(0), or equivalently a Hadamard manifold. A second result is that a Ptolemaic Finsler manifold is necessarily Riemannian. Hence the class of Finsler manifolds does not provide more examples of Ptolemaic spaces. Finally, the authors characterize the Euclidean space among all Riemannian manifolds, using the inversion metrics.
Reviewer: Oana Constantinescu (Iasi)
MSC:
| 53C20 | Global Riemannian geometry, including pinching |
| 53C60 | Global differential geometry of Finsler spaces and generalizations (areal metrics) |
| 51F99 | Metric geometry |
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