Four-dimensional simply connected symplectic symmetric spaces (Q1381335)

The author classifies all four-dimensional simply connected symplectic symmetric spaces. A symmetric space is called symplectic if it is endowed with a symplectic structure which is invariant with respect to the symmetries. Since any symplectic symmetric space has a unique symplectic connection which makes it an affine symmetric space, the study of symplectic symmetric spaces is reduced to a purely algebraic one. From the classification, one finds a class of affine symmetric spaces with a non-abelian solvable transvection group. Any manifold of this class is diffeomorphic to \(\mathbb{R}^{n}\) and has the property that any tensor field on it which is invariant by the transvection group is constant. In particular, its symplectic connection is not a metric one. Moreover, one finds examples of non flat affine symmetric connections on \(\mathbb{R}^{n}\) invariant by the translations and, making quotients, of locally affine symmetric tori which are not globally symmetric.