The Lie group of real analytic diffeomorphisms is not real analytic

Volume 229 / 2015

Rafael Dahmen, Alexander Schmeding Studia Mathematica 229 (2015), 141-172 MSC: Primary 58D15; Secondary 58D05, 22E65, 58B10, 26E05. DOI: 10.4064/sm8130-12-2015 Published online: 21 December 2015


We construct an infinite-dimensional real analytic manifold structure on the space of real analytic mappings from a compact manifold to a locally convex manifold. Here a map is defined to be 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 that this 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.


  • Rafael DahmenFachbereich Mathematik
    Technische Universität Darmstadt
    64289 Darmstadt, Germany
  • Alexander SchmedingInstitutt for matematiske fag
    NTNU Trondheim
    7032 Trondheim, Norway

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