0

In Ludwig Arnold's Random Dynamical Systems, he says

On a paracompact manifold $M$ there is a $C^\infty$ structure subordinate to a $C^k$ structure of $M$ for $1\le k<\infty$ (Whitney).

I don't quite understand his meaning (like, what does 'subordinate to' mean?). I only know Whitney Embedding Theorem, but I hardly see any connections. And I didn't find any other Whitney's theorems related it.

Fan
  • 1,115
  • 1
    This is literally a theorem of Whitney. It is not related to the embedding theorem. It means that there is a subatlas of your $C^k$ structure for which all the transition functions are smooth. –  Aug 08 '16 at 17:39
  • @MikeMiller Thanks a lot for your comment. Could I ask where I can find this theorem and its proof? That would be of great help. – Fan Aug 08 '16 at 17:45
  • There's a proof in Hirsch's differential topology book. –  Aug 08 '16 at 17:53
  • @MikeMiller Thanks :) – Fan Aug 08 '16 at 17:58

0 Answers0