[Paper Review] A non regular Fr\"olicher Lie group of diffeomorphisms

Jean-Pierre Magnot|arXiv (Cornell University)|Jan 12, 2011

Nonlinear Waves and Solitons4 references3 citations

TL;DR

This paper demonstrates that the diffeomorphism group 𝒟 on the open unit interval I, equipped with the topology of uniform convergence of all derivatives on compact sets, is non-regular: the exponential map fails to be defined for certain paths in the Lie algebra. The non-integrable path identified corresponds to a trivial solution of Burger's equation, revealing a deep link between nonlinear PDEs and the geometry of diffeomorphism groups.

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. The non integrable path that we exhibit appears also as a trivial solution of the Burger's equation.

Motivation & Objective

• To investigate the regularity of the diffeomorphism group of the open unit interval I under the Frölicher group topology.
• To determine whether the exponential map is well-defined for all smooth paths in the Lie algebra of 𝒟.
• To identify a specific path in the Lie algebra that fails to be integrable, thus proving non-regularity.
• To explore the connection between such non-integrable paths and solutions of Burger's equation.

Proposed method

• The group 𝒟 of diffeomorphisms on the open unit interval I is endowed with the Frölicher group structure via uniform convergence of all derivatives on compact subsets.
• The Lie algebra of 𝒟 is identified as the space of smooth vector fields on I with appropriate decay or growth conditions.
• A specific path in the Lie algebra is constructed that does not integrate to a one-parameter subgroup, proving non-regularity.
• The path is shown to correspond to a trivial solution of Burger's equation, linking geometric non-regularity to PDE theory.

Experimental results

Research questions

• RQ1 Is the diffeomorphism group of the open unit interval regular under the Frölicher group topology?
• RQ2 Does the exponential map fail to be defined for some smooth paths in the Lie algebra of 𝒟?
• RQ3 Can a non-integrable path in the Lie algebra of 𝒟 be explicitly constructed and linked to a known PDE?
• RQ4 What is the geometric significance of solutions to Burger's equation in the context of infinite-dimensional Lie groups?
• RQ5 How does the failure of integrability relate to the topology of the diffeomorphism group?

Key findings

• The diffeomorphism group 𝒟 on the open unit interval is non-regular, as the exponential map is not defined for certain paths in its Lie algebra.
• A specific non-integrable path in the Lie algebra is explicitly constructed, demonstrating the failure of the exponential map.
• The non-integrable path corresponds to a trivial solution of Burger's equation, establishing a direct link between PDE solutions and geometric non-regularity.
• The topology of uniform convergence of all derivatives on compact sets plays a crucial role in the non-regularity of the group structure.
• The result shows that even in one-dimensional settings, diffeomorphism groups can fail to be regular, challenging assumptions about smoothness in infinite-dimensional Lie theory.