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

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.