On the pseudogroup of local transformations commuting with a transversely elliptic operator and the existence of transverse metric

TL;DR

This paper investigates the pseudogroup of local transformations commuting with a transversely elliptic operator, establishing conditions for equicontinuity and quasi-analyticity, and demonstrating the existence of a transverse metric on the normal bundle.

Contribution

It introduces conditions under which the pseudogroup exhibits equicontinuity and quasi-analyticity, enabling the construction of a transverse metric in foliation theory.

Findings

Pseudogroup of local transformations is equicontinuous under certain conditions.

Quasi-analyticity of the pseudogroup is established with specific operator conditions.

A transverse metric on the normal bundle can be constructed using the Average Method.

Abstract

The group of diffeomorphisms commuting with an elliptic operator on a manifold is a compact Lie group under Compact-Open topology. In foliation theory, pseudogroup is introduced by Sacksteder. The pseudogroup of local transformations commuting with a basic differential operator possesses the equicontinuity and the quasi-analyticity when conditions on operator are given. These properties serve to construct a transverse metric on the normal bundle under a good condition on operator. For this, the Average Method is applied as in the construction of basic connection on foliated bundles.