2007/03/16 by Karl‐Hermann Neeb, Karl-Hermann Neeb, Friedrich Wagemann · 33 citations
Mathematics · #Combinatorics #Diffeomorphism #Differential geometry #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Group (periodic table) #Holomorphic function #Lie algebra #Lie group #Manifold (fluid mechanics) #Mathematics #Physics #Pure mathematics #Simple Lie group #Submanifold #math.DG #msc:22E15 #msc:22E30 #msc:22E65 #msc:22E67
paper · pdf · doi:10.1007/s10711-008-9244-2
published in Geometriae Dedicata 134(1), 17-60 (Springer Science+Business Media) · 39 pages
arxiv created 2007/03/16 · openalex publication_date 2008/03/28 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We study Lie group structures on groups of the form C^∞(M,K), where M is a non-compact smooth manifold and K is a, possibly infinite-dimensional, Lie group. First we prove that there is at most one Lie group structure with Lie algebra C^∞(M,k) for which the evaluation map is smooth. We then prove the existence of such a structure if the universal cover of K is diffeomorphic to a locally convex space and if the image of the left logarithmic derivative in Ω1(M,k) is a smooth submanifold, the latter being the case in particular if M is one-dimensional. We also obtain analogs of these results for the group O(M,K) of holomorphic maps on a complex manifold with values in a complex Lie group. We show that there exists a natural Lie group structure on O(M,K) if K is Banach and M is a non-compact complex curve with finitely generated fundamental group.