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I'm reading a paper where it is shown a topological manifold $N$ has a $C^1$ structure. The very next concept that is expressed is that prior knowledge of an existing homeomorphism $h: M\rightarrow N$ (where $M$ is $C^\infty$) let us now claim it is a "diffeomorphism".

I'm aware that a $C^1$ structure contains an essentially unique $C^\infty$ structure, but definitely wouldn't mind reading someones attempt to clarify this.

My real questions is: What precisely does the author mean when she says "the diffeomorphism $h$" where $h:M\rightarrow N$ where $M$ is $C^\infty$ and $N$ is $C^1$?

Mud
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  • without further information this seems wrong. there definitely exists smooth manifolds $M$ and $N$ which are homeomorphic but not diffeomorphic – Albert Mar 29 '13 at 16:03
  • I'm not asking about the correctness of the statement, rather the definition of a "diffeomorphism" from a C^\infty manifold to a C^1 manifold. – Mud Mar 29 '13 at 16:09

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The definition of a diffeomorphism from a $C^{\infty}$ manifold $M$ to a $C^1$ manifold $N$ is the following. The function $f: M \to N$ is a diffeomorphism if $f$ is a $C^1$ function in local coordinates and if there exists an inverse function that is also $C^1$ in local coordinates. Thus, just use the $C^1$ structure on $N$ and $M$.

treble
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