On the group of $\omega^{k}$-preserving diffeomorphisms

TL;DR

This paper proves that any diffeomorphism of a symplectic manifold preserving a certain power of the symplectic form and connected to the identity must be a symplectomorphism, clarifying the structure of such transformations.

Contribution

It establishes a new characterization of symplectomorphisms based on preservation of ^k forms, extending understanding of symplectic invariance.

Findings

Preserving ^k forms implies being a symplectomorphism if connected to identity.

The result applies for 0 < k < n on symplectic manifolds.

Connectedness to the identity is crucial for the conclusion.

Abstract

We show that if a diffeomorphism of a symplectic manifold $(M^{2n},\omega)$ preserves the form $\omega^{k}$ for $0 < k < n$ and is connected to identity through such diffeomorphisms then it is indeed a symplectomorphism.