The Lie group of real analytic diffeomorphisms is not real analytic

TL;DR

This paper constructs an infinite dimensional real analytic manifold structure for real analytic mappings and investigates the Lie group properties of the diffeomorphism group, showing it is not real analytic in the studied sense.

Contribution

It introduces a manifold structure for real analytic mappings and demonstrates that the Lie group of real analytic diffeomorphisms is not real analytic in the standard sense.

Findings

The diffeomorphism group is a smooth locally convex Lie group.

The group is regular in Milnor's sense.

It is not a real analytic Lie group in the studied framework.

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.