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Dimensional curvature identities in Fedosov geometry

Adrián Gordillo-Merino, Raúl Martínez-Bohórquez, José Navarro-Garmendia

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 Kahler geometry.

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 Kahler geometry.
Paper Structure (9 sections, 9 theorems, 33 equations)

This paper contains 9 sections, 9 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Statements
  3. Dimensional reduction
  4. Main result
  5. Preliminaries
  6. Natural operations on a Fedosov manifold
  7. Invariant theory of the symplectic group
  8. On the Chern forms of symplectic connections
  9. Proof of the general statement

Key Result

Theorem 1.1

The vector space of natural functions, homogeneous of weight $- 4$, that vanish on dimension $2$ is a one dimensional vector space, generated by the function On higher dimensions, there are no homogeneous scalar identities of weight -4.

Theorems & Definitions (19)

  • Theorem 1.1: Scalar identities
  • Theorem 1.2: 2-covariant identities
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Corollary 2.5
  • Conjecture
  • Definition 3.1
  • Theorem 3.2
  • ...and 9 more