An invitation tu quasihomogeneous rigid geometric structures

This is a survey article dealing with quasihomogeneous geometric structures, in the sense that they are locally homogeneous on a nontrivial open set, but not on all of the manifold. Our motivation comes from Gromov's open-dense orbit theorem which asserts that, if the pseudogroup of local automorphisms of a rigid geometric structure acts with a dense orbit, then this orbit is open. Fisher conjectured that the maximal open set of local homogeneity is all of the manifold as soon as the following three conditions are fulfilled: the automorphism group of the manifold acts with a dense orbit, the geometric structure is a $G$-structure (meaning that it is locally homogeneous at the first order) and the manifold is compact. In a recent joint work, with Adolfo Guillot, we succeeded to prove Fisher's conjecture for real analytic torsion free affine connections on surfaces: we construct and classify those connections which are quasihomogenous; their automorphism group never acts with a dense orbit