A construction of symplectic connections through reduction

We give an elementary construction of symplectic connections through reduction. This provides an elegant description of a class of symmetric spaces and gives examples of symplectic connections with Ricci type curvature, which are not locally symmetric; the existence of such symplectic connections was unknown. Key-words: Marsden-Weinstein reduction, symplectic connections, symmetric spaces MSC 2000: 53B05, 53D20, 53C15 1 e-mail: [email protected] 2 Research supported by the Marie Curie Fellowship Nr. HPMF-CT-1999-00062 3 e-mail: [email protected] 4 Research supported by an ARC of the “Communauté française de Belgique” 2 a construction of symplectic connections through reduction 1. Let (M,ω) be a smooth 2n-dimensional symplectic manifold; a linear connection ∇ on (M,ω) is said to be symplectic if it is torsion free and if ω is parallel. If n > 1 the curvature tensor R of such a connection has two irreducible components under the pointwise action of the linear symplectic group Sp(n,R) [1] [2]. We shall denote them by E and W : R = E +W. (1) The E component is determined by the Ricci tensor of ∇; if the W component vanishes the curvature is said to be of Ricci type. In [3] the simply connected symmetric symplectic spaces, whose curvature is of Ricci type have been classified algebraically. It was shown that an isomorphism class was determined by the orbit, under the action of the linear symplectic group Sp(n,R), of an element A belonging to the Lie algebra sp(n,R) of the linear symplectic group such that A = λ Id. (2) where λ is any real number. It was also observed that the only compact symmetric symplectic space with non-zero curvature of Ricci type is the complex projective space Pn(C). In this paper we give a very elementary and geometrical construction of those symmetric symplectic spaces and we provide examples of symplectic connections with Ricci type curvature, which are not locally symmetric. Finally we give a suggestion how to generalize this easy construction. 2. Let A 6= 0 be an element of sp(n + 1,R) and let H be a homogeneous polynomial of degree 2 on R defined by: H(x) = ω(x,Ax), ∀x ∈ R (3) where ω is the standard symplectic structure on R. Let 0 6= μ0 ∈ R and denote by Σμ0 the quadric on R : Σμ0 = {