Construction of Ricci-type connections by reduction and induction
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Given the Euclidean space $\R^{2n+2}$ endowed with a constant symplectic structure and the standard flat connection, and given a polynomial of degree 2 on that space, Baguis and Cahen have defined a reduction procedure which yields a symplectic manifold endowed with a Ricci-type connection. We observe that any symplectic manifold of dimension greater than 2 endowed with a symplectic connection of Ricci-type is locally given by a local version of such a reduction.
We also consider the reverse of this reduction procedure, an induction procedure: we construct globally on a symplectic manifold endowed with a connection of Ricci-type $(M,ω,\nabla)$ a circle or a line bundle which embeds in a flat symplectic manifold $(P,μ,\nabla^1)$ as the zero set of a function whose third covariant derivative vanishes, in such a way that $(M,ω,\nabla)$ is obtained by reduction from $(P,μ,\nabla^1)$.
We further develop the particular case of symmetric symplectic manifolds with Ricci-type connections.