The group of diffeomorphisms of a non compact manifold is not regular

TL;DR

This paper demonstrates that the group of diffeomorphisms on non-compact manifolds, including the open interval, is not regular because the exponential map fails to be defined for certain paths, extending this result to all such manifolds.

Contribution

It proves the non-regularity of diffeomorphism groups on non-compact manifolds, highlighting limitations in their Lie group structure.

Findings

Exponential map is not defined for some paths in the Lie algebra.

The non-regularity extends to all finite-dimensional non-compact manifolds.

Diffeomorphism groups on non-compact manifolds lack certain Lie group properties.

Abstract

We show that a group of diffeomorphisms $\D$ on the open unit interval $I,$ equipped with the topology of uniform convergence on any compact set of the derivatives at any order, is non regular: the exponential map is not defined for some path of the Lie algebra. this result extends to the group of diffeomorphisms of finite dimensional, non compact manifold $M.$