Let $n \geq 2$ and consider two hypersurfaces $H_1$ and $H_2$ of $\mathbb{R}^n$, as well as an application $\phi: \mathbb{R}^n \to \mathbb{R}^n$,such that the restriction of $\phi$ from $H_1$ is bijective with values in $H_2$. What are in general the conditions required for the application $\phi$ viewed as an application from $H_1$ to $H_2$ to be diffeomorphic? Do I need to check that the derivative of $\phi$ does not cancel on the tangent space of any point of $H_1$?
Any reference on this kind of problem would be much appreciated.