Manifolds of mappings on Cartesian products

Abstract

AbstractGiven smooth manifolds $$M_1,\ldots , M_n$$ M 1 , … , M n (which may have a boundary or corners), a smooth manifold N modeled on locally convex spaces and $$\alpha \in ({{\mathbb {N}}}_0\cup \{\infty \})^n$$ α ∈ ( N 0 ∪ { ∞ } ) n , we consider the set $$C^\alpha (M_1\times \cdots \times M_n,N)$$ C α ( M 1 × ⋯ × M n , N ) of all mappings $$f:M_1\times \cdots \times M_n\rightarrow N$$ f : M 1 × ⋯ × M n → N which are $$C^\alpha $$ C α in the sense of Alzaareer. Such mappings admit, simultaneously, continuous iterated directional derivatives of orders $$\le \alpha _j$$ ≤ α j in the jth variable for $$j\in \{1,\ldots , n\}$$ j ∈ { 1 , … , n } , in local charts. We show that $$C^\alpha (M_1\times \cdots \times M_n,N)$$ C α ( M 1 × ⋯ × M n , N ) admits a canonical smooth manifold structure whenever each $$M_j$$ M j is compact and N admits a local addition. The case of non-compact domains is also considered.