## Notes on locally CAT(1) spaces

### B. H. Bowditch

We give an account of complete locally compact locally CAT(1) spaces.
We show that a closed geodesic in such a space cannot be freely homotoped
to a point through rectifiable curves of length strictly less than
$ 2 \pi $.
We deduce that such a space is globally CAT(1) if and only if the space
of closed curves of length less than $ 2 \pi $ is connected.
For these results, we use the Birkhoff curve shortenning process.
We give an example of a smooth riemannian metric on the 3-torus for which
the Birkhoff curve shortenning process fails to converge.
We also describe some area inqualities for surfaces of curvature at most 1.

in ``Geometric group theory'', (ed. R.Charney, M.Davis, M.Shapiro),
de Gruyter (1995) 1-48.

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