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

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. MSC(2010): 22E65; 22E66.

On Enriched Categories and Induced Representations

“…In this paper, we have discussed diffeological Lie groups, and it is natural to ask whether they are regular. J-P. Magnot [Mag17] has shown that this is not the case. So we need to investigate further if regular diffeological Lie groups can be classified.…”

On the geometry of $Diff(S^1)$-pseudodifferential operators based on renormalized traces.

“…One then can consider "natural" notions of smoothness, inherited from the embedding into Cl(S 1 , V ) for the pseudo-differential part, and from the well-known structure of ILB Lie group [31] from the diffeomorphism (phase) component. In order to be more rigorous, one can then consider Frölicher Lie groups along the lines of [23,25] in this context, or in [24,27] when dealing with other examples where this setting is useful. A not-so-complete description of technical properties of Frölicher Lie groups can be found in works by other authors [16,29,30] but this area of knowledge, however, still needs to be further developed.…”

On the differential geometry of numerical schemes and weak solutions of functional equations

“…On a finite dimensional manifold M, the Lie algebra of the ILH Lie group of diffeomorphisms [36], defined as the tangent cone at the identity map, is the space of smooth vector fields, i.e. smooth sections of T M. On a non-compact, locally compact manifold, the situation is quite similar [32] while the group of diffeomorphisms is no longer a Fréchet manifold but a Frölicher Lie group. On these groups, the underlying diffeology is the functional diffeology, as well as for the more general definition of Dif f (X) when X is a diffeological space [21].…”