## 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.