Developability and nonpositive curvature
by
Andre Haefliger
University of Geneva

In the first lecture, which is meant to be elementary, we shall consider the particular case of orbifolds to illustrate the notions of coverings and developability, notions extended later on to the more general case of étale groupoids. We intend to give a complete proof of the fact that an orbifold with a "geometric structure" is always developable.

In the second lecture, the notion of groupoid G of local isometries will be introduced (in the case of Riemannian orbifolds, the underlying groupoid is the groupoid of germs of changes of uniformizing charts), as well as the notions of fundamental group and coverings for G. Such groupoids arise naturally in the theory of Riemannian foliations and in the theory of complexes of groups. We plan to sketch a proof of the following theorem generalizing results of Gromov, Gersten-Stallings, Corson, Spieler, etc.

Theorem Any groupoid of local isometries which is connected, Hausdorff, complete and of non-positive curvature is developable.

This means that such a groupoid is equivalent to the groupoid associated to an action of a group G on a complete simply-connected metric space [X\tilde] of non-positive curvature. The group G will be the fundamental group of the groupoid G and the space [X\tilde] will be the space of G-geodesics issuing from a base point. The proof is an adaptation of the proof of the Cartan-Hadamard theorem given by S. Alexander and R.L. Bishop.

Date received: April 20, 1998