Dimensional curvature identities in Fedosov geometry

Abstract

The curvature tensor of a symplectic connection, as well as its covariant derivatives, satisfy certain identities that hold on any manifold of dimension less than or equal to a fixed n.In this paper, we prove certain results regarding these curvature identities. Our main result describes, for any fixed dimension and any even number p of indices, the first space (provided we have filtered the identities by a homogeneity condition) of p-covariant curvature identities.To this end, we use recent results on the theory of natural operations on Fedosov manifolds. These results allow us to apply the invariant theory of the symplectic group, with a method that is analogous to that used in Riemannian or Kähler geometry.