Lie groups of 𝐶^𝑘-maps on non-compact manifolds and the fundamental theorem for Lie group-valued mappings

Abstract

We study the existence of Lie group structures on groups of the form C k ⁢ ( M , K ) C^{k}(M,K) , where 𝑀 is a non-compact smooth manifold with rough boundary and 𝐾 is a, possibly infinite-dimensional, Lie group. Motivated by introducing this new class of infinite-dimensional Lie groups, we obtain a new version of the fundamental theorem for Lie algebra-valued functions.

Manifolds of mappings on cartesian products

“…was first obtained in [24], for smooth manifolds M j without boundary which are compact or diffeomorphic to R. The Lie group structure for n = 1 was first obtained in [2] for domains diffeomorphic to intervals, together with a sketch for the case n = 2 (assuming additional conditions, e.g. α 1 ≥ r + 3 and…”

“…Our approach differs: While the studies in [24] and [2] assume regularity of G from the start to enforce exponential laws, and build it into a notion of Lie group structures on mapping groups that are "compatible with evaluations," we take canonical and pre-canonical manifold structures as the starting point (independent of regularity) and combine them with regularity or compatibility with evaluations (adapted to C α -maps in Definition 6.2) only when needed.…”

“…(b) Familiar examples of mapping groups turn out to be canonical, notably loop groups C k (S 1 , G) for G a Lie group, and certain Lie groups of the form [2,24]. We extend these constructions in Section 6.…”