Weighted diffeomorphism groups of Riemannian manifolds

TL;DR

This paper develops a framework for weighted diffeomorphism groups on Riemannian manifolds using locally convex vector spaces of weighted vector fields, ensuring inclusion of compactly supported diffeomorphisms and connecting to prior Euclidean space results.

Contribution

It introduces a new class of Lie groups of weighted diffeomorphisms on Riemannian manifolds using locally convex spaces, extending previous Euclidean space constructions.

Findings

Conditions on weights ensure the groups contain compactly supported diffeomorphisms.

The constructed groups coincide with earlier Euclidean space groups in special cases.

Provides a foundation for further analysis of diffeomorphism groups on manifolds.

Abstract

In this paper, we define locally convex vector spaces of weighted vector fields and use them as model spaces for Lie groups of weighted diffeomorphisms on Riemannian manifolds. We prove an easy condition on the weights that ensures that these groups contain the compactly supported diffeomorphisms. We finally show that for the special case where the manifold is the euclidean space, these Lie groups coincide with the ones constructed in the author's earlier work ["Weighted diffeomorphism groups of Banach spaces and weighted mapping groups". In: Dissertationes Math. 484 (2012), p. 128. DOI: 10.4064/dm484-0-1].