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

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

2007-03-15

arxiv.org/abs/math/0703460v2

Geometriae Dedicata 134 (2008) 17--60

Mathematics

Differential Geometry

39 pages

Scientific paper

10.1007/s10711-008-9244-2

We 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 \Omega^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.

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