$C^l$ diffeomorphism between a smooth manifold and a $C^k$ manifold

Question

Let $M$ and $N$ be two Riemannian manifolds. $M$ is smooth while $N$ is $C^k$ manifold.

Suppose there is a $C^l$ diffeomorphism between the two manifolds for $l \leq k$.

Is it true that $N$ is also a smooth manifold?

I am thinking of $M$ and $N$ being hypersurfaces in $\mathbb{R}^n$, for example, like those in these papers: http://arxiv.org/pdf/1106.0622v4.pdf

http://www.igpm.rwth-aachen.de/Download/reports/reusken/ARpaper51.pdf

Answer 1

Not quite. A $C^\infty$-atlas is contained in a $C^k$-atlas, since all $C^\infty$-charts are $C^k$. So, in a sense, the answer is no.

On the other hand, if $N$ is $C^0$, $M$ is $C^\infty$, and $N$ is homeomorphic to $M$, then $N$ admits a smooth structure (Can you see it?) as well.