Manifolds of mappings on cartesian products

TL;DR

This paper constructs a canonical smooth manifold structure on the space of functions with specified smoothness on products of manifolds, extending to non-compact cases, with applications in differential geometry.

Contribution

It establishes a new manifold structure on mapping spaces on product manifolds, including non-compact cases, generalizing previous results.

Findings

Manifold structure exists for mappings on compact product manifolds.

Extension of manifold structure to non-compact domains.

Applicability to manifolds modeled on locally convex spaces.

Abstract

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 $f\colon M_1\times\cdots\times M_n\to N$ which are $C^\alpha$ in the sense of Alzaareer. Such mappings admit, simultaneously, continuous iterated directional derivatives of orders $\leq \alpha_j$ 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.