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.