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The Banach manifold \(C^{k}(M,N)\). (English) Zbl 1433.58012

Let \(M\) be a closed smooth manifold and \(N\) a connected, smooth finite-dimensional manifold without boundary. The author gives the rigorous, complete proof of the well known result that for each \(k \in \mathbb{N}\) the set \(C^k(M,N)\) of \(k\) times continuously differentiable mappings is a smooth Banach manifold. The author proves a version of the so-called global \(\Omega\)-lemma which is the key point in constructing a smooth Banach manifold structure on \(C^k(M,N)\). In addition, it is proved that the manifold topology of \(C^k(M,N)\) is the compact open \(C^k\) topology.

MSC:

58D15 Manifolds of mappings
57N20 Topology of infinite-dimensional manifolds
58B05 Homotopy and topological questions for infinite-dimensional manifolds
58B10 Differentiability questions for infinite-dimensional manifolds
58C25 Differentiable maps on manifolds
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References:

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