Manifold of smooth mappings

Summary

Let $M$ be a compact manifold and $N$ a manifold. The set $C^\infty(M, N)$ of all smooth maps is itself an infinite-dimensional manifold, modelled on Fréchet spaces of smooth sections. The charts come from the exponential map of a Riemannian metric on $N$. A map near $f$ is written as $x \mapsto \exp_{f(x)}(s(x))$ for a small vector field $s$ along $f$. Loop spaces, diffeomorphism groups and spaces of embeddings are open subsets or submanifolds of such manifolds. Maps of class $C^k$, or of Sobolev class $H^s$ with $s > \dim M / 2$, form Banach or Hilbert manifolds of maps, which are better suited to analysis.