Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds

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dc.creatorNeeb, Karl-Hermann
dc.creatorWagemann, Friedrich
dc.date2007-03-15
dc.date2007-03-16
dc.date.accessioned2026-07-07T10:00:28Z
dc.date.available2026-07-07T10:00:28Z
dc.descriptionWe study Lie group structures on groups of the form C^\infty(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^\infty(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 Ω^1(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.
dc.description39 pages
dc.identifierhttps://arxiv.org/abs/math/0703460
dc.identifierhttp://arxiv.org/abs/math/0703460
dc.identifierGeometriae Dedicata 134 (2008) 17--60
dc.identifierdoi:10.1007/s10711-008-9244-2
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/168329
dc.subjectDifferential Geometry
dc.subject22E65, 22E67, 22E15, 22E30
dc.titleLie group structures on groups of smooth and holomorphic maps on non-compact manifolds
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