Appendix C - Canonical Manifold of Mappings

Summary

This appendix sketches the construction of a canonical manifold of mappings structure for smooth mappings between (finite-dimensional) manifolds. Before we begin, let us consider for a moment the locally convex space of smooth functions from a manifold with values in a locally convex space. The topology and vector space structure allow us to compare two smooth maps by measuring their pointwise difference on compact sets. As manifolds lack an addition we can not mimick this for manifold valued functions (albeit the topology still makes sense!). On first sight, it might be tempting to think that one could use the charts of the target manifold to construct charts for the smooth functions. However, if the target manifold does not admit an atlas with only one chart, there will be smooth mappings whose image is not contained in one chart. Thus the charts of the target manifold turn out to be not very useful. Instead one needs to find a replacement of the vector space addition to construct a way in which charts vary smoothly over the target manifold. This leads to the concept of a local addition which enables the construction of a manifold structure.