Let $f:\mathbb{R}^{n-1}\rightarrow\mathbb{R}$ be a smooth map, and $F:\mathbb{R}^n\rightarrow\mathbb{R}$ be defined as $F(x_1,\cdots,x_n)=f(x_1,\cdots,x_{n-1})-x_n$. How to prove that $F^{-1}(0)$ is diffeomorphic to $\mathbb{R}^{n-1}$?
Actually, I can prove that $F^{-1}(0) $ is a regular level set and thus a $n-1$ dimension submanifold of $\mathbb{R}^n$. Can I conclude directly that $F^{-1}(0)$ is diffeomorphic to $\mathbb{R}^{n-1}$?
Or more generally, I want to prove $SU(2)$ is diffeomorphic to $S^3$. It follows directly that $SU(2)$ is a Lie group of dimension $3$. Is this enough to show the diffeomorphism?
Thanks for any help!
You just need to show that $$\begin{align*}g:&\,F^{-1}(0)=\{(x_1,...,x_{n-1},x_n):f(x_1,..,x_{n-1})=x_n\}\rightarrow \mathbb{R}^{n-1}\\&(x_1,...,x_{n-1},x_n)\mapsto (x_1,...,x_{n-1}) \end{align*}$$ is a diffeomorphism.