Symplectic Connections Induced by the Chern Connection

TL;DR

This paper explores how the Chern connection of a Finsler structure on a symplectic manifold can induce symplectic connections, leading to a family of Fedosov structures under certain conditions.

Contribution

It introduces a method to lift the symplectic form to the tangent bundle and characterizes when the Chern connection preserves this lift, resulting in new Fedosov structures.

Findings

Conditions for the Chern connection to preserve the lifted symplectic form

Existence of a family of Fedosov structures on the manifold

Relation between vector fields and Fedosov structures

Abstract

Let $(M,\omega)$ be a symplectic manifold and $F$ be a Finsler structure on $M$. In the present paper we define a lift of the symplectic two-form $\omega$ on the manifold $TM\backslash 0$, and find the conditions that the Chern connection of the Finsler structure $F$ preserves this lift of $\omega$. In this situation if $M$ admits a nowhere zero vector field then we have a non-empty family of Fedosov structures on $M$.