Four-dimensional simply connected symplectic symmetric spaces

A symplectic symmetric space is a symmetric space endowed with a symplectic structure which is invariant by the symmetries. We give here a classification of four-dimensional symplectic symmetric spaces which are simply connected. This classification reveals a remarkable class of affine symmetric spaces with a non-Abelian solvable transvection group. The underlying manifold M of each element (M, del) belonging to this class is diffeomorphic to R-n with the property that every tensor field on M invariant by the transvection group is constant; in particular, del is not a metric connection. This classification also provides examples of nonflat affine symmetric connections on R-n which are invariant under the translations. By considering quotient spaces, one finds examples of locally affine symmetric tori which are not globally symmetric.