The Banach manifold $C^k(M,N)$

TL;DR

This paper rigorously proves that the set of $k$-times differentiable maps between a closed manifold and a boundaryless connected manifold forms a smooth Banach manifold under the compact-open $C^k$ topology.

Contribution

It provides a detailed and rigorous proof of the Banach manifold structure on $C^k(M,N)$, clarifying and completing existing partial results.

Findings

Establishes the Banach manifold structure for $C^k(M,N)$

Provides detailed proof filling gaps in previous literature

Confirms the smoothness of the manifold structure

Abstract

Let $M$ be a closed manifold and let $N$ be a connected manifold without boundary. For each $k\in\mathbb{N}$ the set of $k$ times continuously differentiable maps between $M$ and $N$ has the structure of a smooth Banach manifold where the underlying manifold topology is the compact-open $C^k$ topology. We provide a detailed and rigorous proof for this important statement which is already partially covered by existing literature.