Differentiable structures on function spaces

Abstract:

A ${C^s}$ differentiable manifold structure is constructed for spaces of maps from a compact ${C^r}$ manifold $M$ to a ${C^{r + s}}$ manifold $N$. The method (1) is inspired by Douady; (2) does not require any additional structure on $N$ (such as sprays); (3) includes the case when $N$ is an analytic manifold and concludes that the mapping space is also an analytic manifold; (4) can be used to treat all the classical mapping spaces (${C^r}$ functions, ${C^r}$ functions with Hölder conditions, and Sobolev functions). Several interesting aspects of these manifolds are investigated such as their tangent spaces, their behavior with respect to functions, and realizations of Lie group structures on them. Differentiable structures are also exhibited for spaces of compact maps with noncompact domain.