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Four-Dimensional Simply Connected Symplectic Symmetric Spaces

par Bieliavsky, Pierre
Référence Geometriae dedicata, 69, 3, page (291-316)
Publication Publié, 1998

Article révisé par les pairs

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Résumé :

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, ∇) belonging to this class is diffeomorphic to Rn with 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 Rn which are invariant under the translations. By considering quotient spaces, one finds examples of locally affine symmetric tori which are not globally symmetric.