Prove that $\mathbb{R}^2 \times S^1 $ and $M=\left\lbrace (x,y,z,w) \in \mathbb{R}^4 \mid x^2+y^2- z^2-w^2 = 1 \right\rbrace$ are diffeomorphic

Question

Let be $M= \left\lbrace (x,y,z,w) \in \mathbb{R}^4 \mid x^2+y^2- z^2-w^2 = 1 \right\rbrace$.

I have proved that $M$ is a embedded submanifold of $\mathbb{R}^4 $ of dimension $3$.

I have now to prove that $M$ is diffeomorphic to $\mathbb{R}^2 \times S^1 $.

Can someone give me an hint to find a diffeomorphism?

Thanks a lot!

Answer 1

HINT: Can you write down a diffeomorphism $$S^1\times\Bbb R\to \{(x,y,z)\in\Bbb R^3: x^2+y^2-z^2=1\}?$$