2009 DG-MP-PDE Seminar - 05

Starting from the standard contact structure of the supercircle, S^{1|1}, one considers the subgroups E(1|1), Aff(1|1), and SpO(2|1) of the group, K(1), of its contactomorphisms that respectively define its Euclidean, affine, and projective geometries. The notion of p|q-transitivity allows one to systematically construct the characteristic invariants of each geometry, in particular the super cross-ratio. One deduces the nontrivial associated 1-cocycles of K(1), e.g., the superschwarzian. The case of the supercircle S^{1|2} is also studied. The aim of this talk is to present in a synthetic fashion these geometric objects which are somewhat scattered in the literature. This is joint work with J.-P. Michel.