Manifolds of mappings associated with real-valued function spaces and natural mappings between them

TL;DR

This paper develops a smooth manifold structure for spaces of continuous mappings from a compact manifold with corners to a finite-dimensional manifold, focusing on natural mappings and superposition operators.

Contribution

It introduces a general framework for manifold structures on mapping spaces based on local addition and simple axioms, extending previous concepts to broader classes of function spaces.

Findings

Constructed smooth manifold structures for mapping spaces.

Analyzed smoothness of natural mappings like superposition operators.

Provided conditions under which these spaces are well-behaved.

Abstract

Let $M$ be a compact smooth manifold with corners and $N$ be a finite dimensional smooth manifold without boundary which admits local addition. We define a smooth manifold structure to general sets of continuous mapings $\mathcal{F}(M,N)$ whenever functions spaces $\mathcal{F}(U,\mathbb{R})$ on open subsets $U\subseteq [0,\infty)^n$ are given, subject to simple axioms. Construction and properties of spaces of sections and smoothness of natural mappings between spaces $\mathcal{F}(M,N)$ are discussed, like superposition operators $\mathcal{F}(M,f):\mathcal{F}(M,N_1)\to \mathcal{F}(M,N_2)$, $\eta \mapsto f\circ \eta$ for smooth maps $f:N_1\to N_2$.