Manifolds of mappings associated with real-valued function spaces and natural mappings between them
Matthieu F. Pinaud
TL;DR
This work develops an axiomatic framework to endow mapping spaces $\mathcal{F}(M,N)$ with smooth (and analytic) infinite-dimensional manifold structures, even when $M$ has corners and the target $N$ admits a local addition. By leveraging a good collection of open sets $\mathcal{U}$ and a family of real-valued function spaces $\mathcal{F}(U,\mathbb{R})$, it constructs $\mathcal{F}(M,N)$ and the associated spaces of $\mathcal{F}$-sections, proving smoothness of natural operations such as the superposition operator $\mathcal{F}(M,f)$ for smooth $f$. A key technical contribution is the identification of the tangent bundle with $\mathcal{F}(M,TN)$ via $\Theta_N$, together with charts built from local additions, ensuring independence from the chosen addition. The paper also provides a concrete Hölder-space example, showing $C^{0,\lambda}(M,N)$ forms a smooth manifold and that composition is smooth, thereby connecting abstract theory to tangible function spaces and Lie-group targets. Overall, the framework extends smooth calculus on mapping spaces to settings with corners and broad classes of function spaces, with potential impact on analysis of nonlinear operators like Nemytskij/ superposition operators on complex domains.
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 $\mathcal{F}(M,f):\mathcal{F}(M,N_1)\to \mathcal{F}(M,N_2)$, $η\mapsto f\circ η$ for smooth maps $f:N_1\to N_2$.