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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