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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm8130-12-2015,
author = {Rafael Dahmen and Alexander Schmeding},
title = {The Lie group of real analytic diffeomorphisms is not real analytic},
journal = {Studia Mathematica},
volume = {231},
year = {2015},
pages = {141-172},
zbl = {1341.58011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm8130-12-2015}
}
Rafael Dahmen; Alexander Schmeding. The Lie group of real analytic diffeomorphisms is not real analytic. Studia Mathematica, Tome 231 (2015) pp. 141-172. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm8130-12-2015/