The Lie group of real analytic diffeomorphisms is not real analytic

Rafael Dahmen ¹   ; Alexander Schmeding ²

¹ Fachbereich Mathematik Technische Universität Darmstadt 64289 Darmstadt, Germany
² Institutt for matematiske fag NTNU Trondheim 7032 Trondheim, Norway

Studia Mathematica, Tome 229 (2015) no. 2, pp. 141-172

Résumé

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.

Détail

DOI : 10.4064/sm8130-12-2015

Keywords: construct infinite dimensional real analytic manifold structure space real analytic mappings compact manifold locally convex manifold here map defined real analytic extends holomorphic map neighbourhood complexification its domain known construction turns group real analytic diffeomorphisms smooth locally convex lie group prove group regular sense milnor inequivalent convenient setting calculus real analytic diffeomorphisms even form real analytic lie group however prove lie group structure group real analytic diffeomorphisms general real analytic sense