Example of a homeomorphism that is not a diffeomorphism between two diffeomorphic surfaces in $\mathbb{R}^3$

Question

Give an example of a homeomorphism that is not a diffeomorphism between two diffeomorphic surfaces in $\mathbb{R}^3$.

The only function that came in my mind is $(x,y) \mapsto (x^3, y^3, \sqrt{1-x^6-y^6})$ between the two surfaces $\{(x,y) \mid x^6 + y^6 < 1\}$ and the image of the function (that I think is $S^2 \cap \{z > 0\}$). Some help? Thanks.

Answer 1

You can take the surfaces $S=S^*=\{(x,y,0)\mid x,y\in\Bbb R\}$. And then consider$$\begin{array}{ccc}S&\longrightarrow&S^*\\(x,y,0)&\mapsto&(x^3,y,0),\end{array}$$which is a homeomorphism, but not a diffeomorphism.