Manifolds of mappings on cartesian products

Abstract

Given smooth manifolds $M_1,\dots ,M_n$ (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 $f:M_1\times \cdots \times M_n\to N$ 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 $j\in \{1,\dots ,n\}$, 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.

48 pages, LaTeX. v2: update of references, minor corrections (all results unchanged)

References in corpus (1)

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

Cited by in corpus (1)

• An introduction to infinite-dimensional differential geometry