I came across the following question whilst preparing for an exam.
Consider the map $f:\mathbb{R}^{3} \to \mathbb{R}$ given by the polynomial $f(x,y,z)=x^{2}+y^{2}+z^{2}$. Let $s,t \neq 0$. Show that the regular surfaces $f^{-1}(t)$ and $f^{-1}(s)$ are diffeomorphic if and only if $t/s$ is positive.
I showed that $f^{-1}(a)$ is a regular surface if $a \neq 0$ using the Regular Value Theorem, but I'm not sure how to show the statement. For the only if direction I had in mind that given $t/s$ is positive we have $t,s>0$ or $t,s<0$ and then construct an explicit diffeomorphism, but I feel like there should be a better way to do this.