On properly essential classical conformal diffeomorphism groups

TL;DR

This paper proves that certain classical conformal diffeomorphism groups are properly essential using a cohomological criterion and explores the orbit structure of tensor fields under these groups on contact manifolds.

Contribution

It establishes the proper essentiality of classical conformal diffeomorphism groups via a new local cohomological criterion and analyzes their action on tensor fields.

Findings

Classical conformal diffeomorphism groups are properly essential.

A local cohomological criterion characterizes conformal diffeomorphisms.

Existence of non-diffeomorphic conformal contact forms on closed contact manifolds.

Abstract

We prove that various classical conformal diffeomorphism groups, which are known to be essential [1], are in fact properly essential. This is a consequence of a local criterion on a conformal diffeomorphism in the form of a cohomological equation. Furthermore, we study the orbit of a tensor field under the action of the conformal diffeomorphism group for these classical conformal structures. On every closed contact manifold, we find conformal contact forms that are not diffeomorphic.