On Rigidity of Generalized Conformal Structures

TL;DR

This paper extends the classical Liouville Theorem to generalized conformal structures, demonstrating that such structures exhibit 2-rigidity, meaning they are uniquely determined by their 2-jet at any point.

Contribution

It introduces a new rigidity result for generalized conformal structures, broadening the understanding of conformal transformation uniqueness beyond classical cases.

Findings

Generalized conformal structures are 2-rigid, determined by their 2-jet at any point.

The work extends classical conformal rigidity results to a broader class of structures.

Provides a foundational result for the study of transformations in generalized geometric frameworks.

Abstract

The classical Liouville Theorem on conformal transformations determines local conformal transformations on the Euclidean space of dimension $\geq 3$. Its natural adaptation to the general framework of Riemannian structures is the 2-rigidity of conformal transformations, that is such a transformation is fully determined by its 2-jet at any point. We prove here a similar rigidity for generalized conformal structures defined by giving a one parameter family of metrics (instead of scalar multiples of a given one) on each tangent space.