A nonstandard operator for almost Grassmannian geometries with a torsion

TL;DR

This paper introduces a new fourth-order invariant differential operator for almost Grassmannian geometries with torsion, extending concepts from conformal geometry to a broader geometric setting.

Contribution

It constructs a novel invariant operator using the curved translation principle, generalizing the Paneitz operator to almost Grassmannian structures with torsion.

Findings

The operator is explicitly constructed via a non-normal tractor connection.

It generalizes the Paneitz operator from conformal to Grassmannian geometries.

The method applies to geometries with arbitrary torsion.

Abstract

A nonstandard invariant fourth order operator acting on functions on a manifold equipped with an almost Grassmannian structure with an arbitrary trorsion is found by means of the curved translation principle. This operator can be viewed as a Grassmannian analogue of the Paneitz operator well known from conformal geometry. It is obtained by translating a Grassmannian analogue of the Laplace operator to a certain tractor bundle with a specific tractor connection, which is not normal in general.