0

I know that a smooth manifold is a topological manifold whose transition maps are smooth.

Must the coordinate maps also be smooth? Must they be diffeomorphisms?

MathWorld seems to think so, but I do not understand why it follows from the definition that this must be the case.

Open Season
  • 1,332
  • 3
    What does it mean for a coordinate map to be smooth? – Tim kinsella Sep 27 '15 at 19:18
  • I'm not sure, I guess this might be the problem. – Open Season Sep 27 '15 at 19:21
  • 1
    Right. There's no intrinsic notion of a derivative of a function on a manifold without resorting to coordinate maps. So we say a map between manifolds $f:M\rightarrow N$ is smooth if the map gotten by pre and post composing f with coordinate maps is smooth. The latter is a map between Euclidean spaces, where the notion of smoothness is familiar. So it's sort of tautological that coordinate maps are smooth because if you pre and post compose them with a coordinate map, taking the former coordinate map in both cases, you get the identity, which is certainly smooth. – Tim kinsella Sep 27 '15 at 19:27
  • 1
    To see that it follows from the definition, make sure you're looking at the relevant definition, not the definition of "smooth manifold" but the definition of "smooth map on a manifold". – Andreas Blass Sep 27 '15 at 19:27
  • I see now, thanks guys. – Open Season Sep 27 '15 at 19:32

0 Answers0