Manifolds of mappings on Cartesian products. (English) Zbl 1484.58005
Summary: Given smooth manifolds \(M_1,\ldots, M_n\) (which may have a boundary or corners), a smooth manifold \(N\) modeled on locally convex spaces and \(\alpha \in ({{\mathbb{N}}}_0\cup \{\infty \})^n\), we consider the set \(C^\alpha (M_1\times \cdots \times M_n,N)\) of all mappings \(f:M_1\times \cdots \times M_n\rightarrow N\) which are \(C^\alpha\) in the sense of Alzaareer. Such mappings admit, simultaneously, continuous iterated directional derivatives of orders \(\le \alpha_j\) in the \(j\) th variable for \(j\in \{1,\ldots , n\} \), in local charts. We show that \(C^\alpha (M_1\times \cdots \times M_n,N)\) admits a canonical smooth manifold structure whenever each \(M_j\) is compact and \(N\) admits a local addition. The case of non-compact domains is also considered.
MSC:
| 58D15 | Manifolds of mappings |
| 22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |
| 26E15 | Calculus of functions on infinite-dimensional spaces |
| 26E20 | Calculus of functions taking values in infinite-dimensional spaces |
| 46E40 | Spaces of vector- and operator-valued functions |
| 46T20 | Continuous and differentiable maps in nonlinear functional analysis |
| 58A05 | Differentiable manifolds, foundations |
Keywords:
infinite-dimensional manifold; infinite-dimensional Lie group; compact-open topology; exponential law; evaluation map; mapping group; regularity; box product; non-compact manifoldReferences:
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