Four-Dimensional Simply Connected Symplectic Symmetric Spaces

A symplectic is a symmetric space endowed with a symplectic structure which is invariant by the symmetries. We give here a classification of four-dimensional symplectic 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, ▿) belonging to this class is diffeomorphic to Rnwith the property that every tensor field on M invariant by the transvection group is constant; in particular, ▿ is not a metric connection. This classification also provides examples of nonflat affine symmetric connections on Rnwhich are invariant under the translations. By considering quotient spaces, one finds examples of locally affine symmetric tori which are not globally symmetric.