Construction of Ricci-type connections by reduction and induction

Abstract

Given the Euclidean space ℝ²ⁿ⁺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 [1] have defined a reduction procedure which yields a symplectic manifold endowed with a Ricci-type connection. We observe that any symplectic manifold (M, ω) of dimension 2n (n ≥ 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, ∇) a circle or a real line bundle which embeds in a flat symplectic manifold (P, μ, ∇¹) as the zero set of a function whose third covariant derivative vanishes in such a way that (M, ω, ∇) is obtained by reduction from (P, μ, ∇¹).

We further develop the particular case of symmetric symplectic manifolds with Ricci-type connections.

This research was partially supported by an Action de Recherche Concertée de la Communauté française de Belgique.