On invariants of almost symplectic connections

TL;DR

This paper analyzes the structure of torsion tensors in almost symplectic connections, classifies symplectic invariants, and provides insights into selecting preferred connections, rederiving known theorems and exploring torsion properties.

Contribution

It offers a detailed decomposition of torsion tensors under Sp(2n, R) and describes symplectic quadratic invariants, advancing understanding of almost symplectic connections.

Findings

Decomposition of torsion tensors under Sp(2n, R)

Description of symplectic quadratic invariants

Insights into properties of vectorial torsion

Abstract

We study the irreducible decomposition under Sp(2n, R) of the space of torsion tensors of almost symplectic connections. Then a description of all symplectic quadratic invariants of torsion-like tensors is given. When applied to a manifold M with an almost symplectic structure, these instruments give preliminary insight for finding a preferred linear almost symplectic connection on M . We rediscover Ph. Tondeur's Theorem on almost symplectic connections. Properties of torsion of the vectorial kind are deduced.