The Lie group of real analytic diffeomorphisms is not real analytic

Abstract

We construct an infinite dimensional real analytic manifold structure for the space of real analytic mappings from a compact manifold to a locally convex manifold. Here a map is real analytic if it extends to a holomorphic map on some neighbourhood of the complexification of its domain. As is well known the construction turns the group of real analytic diffeomorphisms into a smooth locally convex Lie group. We prove then that the diffeomorphism group is regular in the sense of Milnor. In the inequivalent "convenient setting of calculus" the real analytic diffeomorphisms even form a real analytic Lie group. However, we prove that the Lie group structure on the group of real analytic diffeomorphisms is in general not real analytic in our sense.

33 pages, LaTex, v2: now includes a proof for the regularity of the real analytic diffeomorphism group

References in corpus (3)

• Lie groups over non-discrete topological fields
• Differentiable mappings on products with different degrees of differentiability in the two factors
• Complexifications of infinite-dimensional manifolds and new constructions of infinite-dimensional Lie groups

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• Group Contractions via Infinite-Dimensional Lie Theory
• A topological splitting of the space of meromorphic germs in several variables and continuous evaluators
• Real analyticity of composition is shy