Symplectic structures on the tangent bundle of a smooth manifold

TL;DR

This paper introduces a method to construct symplectic forms on the tangent bundle of a manifold from tensor fields and characterizes all such forms near the zero section as naturally derived from these tensors.

Contribution

It provides a systematic way to lift tensor fields to symplectic forms on tangent bundles and classifies symplectic forms near the zero section in terms of these tensors.

Findings

A method to lift (2,0)-tensor fields to symplectic forms on TM.

Any symplectic form on TM near the zero section is symplectomorphic to one built from three tensor fields.

The classification of symplectic forms on TM in a neighborhood of the zero section.

Abstract

We give a method to lift $(2,0)$-tensors fields on a manifold $M$ to build symplectic forms on $TM$. Conversely, we show that any symplectic form $\Om$ on $TM$ is symplectomorphic, in a neighborhood of the zero section, to a symplectic form built naturally from three $(2,0)$-tensor fields associated to $\Om$.