Infinitesimal affine automorphisms of symplectic connections

TL;DR

This paper investigates conditions under which infinitesimal automorphisms of symplectic connections are symplectic vector fields, providing examples where this does not hold, thus deepening understanding of symplectic geometry.

Contribution

It establishes criteria for infinitesimal automorphisms to be symplectic vector fields and presents a counterexample on a compact symplectic manifold.

Findings

Conditions for automorphisms to be symplectic vector fields

Existence of non-symplectic infinitesimal automorphisms on flat symplectic manifolds

Counterexample on a compact symplectic manifold

Abstract

Conditions are given under which an infinitesimal automorphism of a torsion-free connection preserving a symplectic form is necessarily a symplectic vector field. An example is given of a compact symplectic manifold admitting a flat symplectic connection and an infinitesimal automorphism that is not symplectic.