? W H A T I S . . . a Projective Structure ?

The theory of locally homogeneous geometric structures on manifolds is a rich playground of examples on the border of topology and geometry. While geometry concerns quantitative relationships between collections of points, topology concerns the loose qualitative organization of points. Given a geometry (such as Euclidean geometry) and a manifold with some topology (such as the round 2-sphere), how many ways can one put the geometry, at least locally, on the manifold? The familiar fact that no metrically accurate world atlas exists is just the fact that the sphere admits no Euclidean geometry. However, the wide variety of geometries (homogeneous spaces of Lie groups) and manifolds leads to a fascinating array ofquestions. Here is a precise definition. Consider a homogeneous space X with a transitive Lie group G of diffeomorphisms. In the spirit of FelixKlein’s 1872 Erlangen program, X admits a geometry defined by thesymmetrygroupG. Klein simply defined the “geometry” to be all the objectsonX togetherwith theG-invariant relations between them. So a Euclidean structure on a manifold is simply a system of Euclidean coordinates related by isometries on overlapping coordinate patches. Such a structure defines aRiemannianmetric locally isometric to Euclidean space (and hence having zero curvature). In fact this structure is equivalent toaflatRiemannianmetric. A projective structure on a manifold M is a system of local coordinates modeled on a projective space P so that on any two overlapping coordinate