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Hyperbolic manifolds have constant sectional curvature $-1$. The two-holed torus, for example, can (I believe) be given a hyperbolic metric so that it has curvature $-1$. It should also be a complete metric space, with this metric.

It would seem as though this should make the two-holed torus with this metric a CAT(0) space (even a CAT(-1) space). However, the two-holed torus is obviously not contractible, but CAT(0) spaces are contractible.

I'm sure I'm just confused about the definitions, but I'm not sure where.

vukov
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    We need a simply connected condition. – HK Lee Nov 29 '17 at 02:19
  • You mean that it follows from the definition of CAT(0) space that it is simply connected? The definition (on wikipedia) doesn't explicitly say that CAT(0) spaces are simply connected. – vukov Nov 29 '17 at 02:30
  • Where do you find "CAT[0] space is contractible" ? – HK Lee Nov 29 '17 at 02:38
  • On https://en.wikipedia.org/wiki/Hadamard_space, "In a Hadamard space, any two points can be joined by a unique geodesic between them; in particular, it is contractible." – vukov Nov 29 '17 at 02:55

1 Answers1

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These manifolds are locally CAT(-1) (and, hence, locally CAT(0)). Note that locally CAT(k) spaces are also said to have curvature $\le k$. To make them globally CAT(0) you need to add "complete and simply connected". ("Complete" will be automatic if your manifold is compact.)

Moishe Kohan
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  • So the definition of CAT(0) (in terms of comparison triangles) implies being simply connected? It isn't explicitly part of the definition? – vukov Nov 29 '17 at 03:07
  • @vukov: No, simple connectivity is not a part of the definition (just read the definition!). However, CAT(0) spaces are contractible (it is a relatively easy theorem). – Moishe Kohan Nov 29 '17 at 13:11