Riemannian metrics and Laplacians for generalised smooth distributions

TL;DR

This paper demonstrates that any generalized smooth distribution on a manifold can be equipped with a Riemannian metric, enabling the construction of a Laplacian that exhibits desirable analytical properties like self-adjointness and hypoellipticity.

Contribution

It introduces a method to define Riemannian metrics and Laplacians for generalised smooth distributions, including those of non-constant rank, and analyzes their analytical properties.

Findings

Existence of Riemannian metrics for all generalized smooth distributions.

Construction of a Laplacian operator with self-adjointness on compact manifolds.

Proof of hypoellipticity of the Laplacian within the pseudodifferential calculus.

Abstract

We show that any generalised smooth distribution on a smooth manifold, possibly of non-constant rank, admits a Riemannian metric. Using such a metric, we attach a Laplace operator to any smooth distribution as such. When the underlying manifold is compact, we show that it is essentially self-adjoint. Viewing this Laplacian in the longitudinal pseudodifferential calculus of the smallest singular foliation which includes the distribution, we prove hypoellipticity.