arXiv:2109.04005v4 [math.DG] 28 Sep 2021 ON THE PSEUDOGROUP OF LOCAL TRANSFORMATIONS COMMUTING WITH A TRANSVERSELY ELLIPTIC OPERATOR AND THE EXISTENCE OF TRANSVERSE METRIC WENRAN LIU Abstract. The group of diffeomorphisms commuting with an elliptic operator on a manifold is a compact Lie group under the Compact-Open topology. In fo- liation theory, pseudogroups were introduced by Sacksteder. The pseudogroup of local transformations commuting with a basic differential operator possesses the equicontinuity and the quasi-analyticity properties when conditions on the opera- tor are given. These properties serve to construct a transverse metric on the normal bundle under a good condition on the operator. For this, the Average Method is applied as in the construction of basic connections on foliated bundles. Contents 1. Introduction 1 2. Prelimiaries 4 2.1. Pseudogroups of transformation 4 2.2. Foliated manifold 5 2.3. Differential operators on foliated manifold 5 3. Commuting Pseudogroup 7 3.1. Equicontinuity of G (P ) 8 3.2. Quasi-Analyticity of G (P ) 10 4. Construction of transverse metric 13 4.1. Closure of sub-pseudogroups of G (P ) 14 4.2. Transitivity of Pseudogroup 15 4.3. Construction of transverse metric 16 References 19 1. Introduction The notion of bundle-like metric on foliated manifold was first introduced by Rein- hart [35]. A foliated manifold equipped with a bundle-like metric is called a Riemann- ian foliation. An equivalent definition is the existence of a transverse (holonomy- invariant) metric on the normal bundle of the foliation. In [32, 31], Molino proposed a theory for the study of Riemannian foliation. In [10] and in [9], Carrière and Breuillard-Gelander studied the growth of Riemannian foliations. In [21], El.Kacimi- Alaoui introduced the notion of transversely elliptic operator and proved the associa- tion of a transversely elliptic operator over a Riemannian foliation and an equivariant Key words and phrases. foliation, pseudogroup, transversely elliptic operator, equicontinuity, quasi-analyticity, transverse metric. This research project is supported by the Starting Research Fund from Civil Aviation University of China, Project No. 2020KYQD58. 1