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Recently, I was reading a paper about the rigidity of negatively curved cone surfaces written by S. Hersonsky and F. Paulin. The authors said that a negatively curved cone surface is locally CAT(-1). But I don't know why.

A negatively curved cone surface is a surface $M$ endowed with a negatively curved cone metric, i.e. a smooth negatively curved Riemannian metric on $M-P$, where $P$ is a discrete subset of points on $M$, such that the completion of $M-P$ is $M$, and such that the cone angle at each singularity is greater than $2\pi$.

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    $\begingroup$ I assume that the metric has curvature less than $-1$ away from the cone points? $\endgroup$
    – Igor Rivin
    Jul 14, 2015 at 16:12
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    $\begingroup$ Note that if you cut a triangle into two thin triangles then it is thin and use it. See 9.3.4 here math.psu.edu/petrunin/papers/alexandrov-geometry $\endgroup$ Jul 14, 2015 at 20:03
  • $\begingroup$ @IgorRivin: the curvature is only strictly negative away from the cone points $\endgroup$ Jul 15, 2015 at 0:47
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    $\begingroup$ If the curvature is only assumed negative away from the cone points, then the result is false, since a Riemannian metric of curvature $-1/2$ is not CAT(-1). $\endgroup$
    – Igor Rivin
    Jul 15, 2015 at 1:19
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    $\begingroup$ @HuipingPan, I gave a ref, and yet for sure you have to assume that curvature $\le -1$. $\endgroup$ Jul 15, 2015 at 14:33

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