Q:

A diffeomorphism between a cylinder and a one-sheeted hyperboloid

A cylinder $C=\{(x,y)\in \mathbb{R}^{3}: x^2 +y^2=1 \}$ and a hyperboloid $H=\{(x,y,z)\in \mathbb{R}^{3}: x^2 +y^2 - z^2=1 \}$, my try was to define the functions $F:O\rightarrow H$ and $G:H \rightarrow O$ as
$F(x,y,z)=(-x,-y,\sqrt{2(x^2+y^2)})$ and $G(x,y,z)=(x,y,0)$ and so conclude that $O$ and $G$ are diffeomorphic.
is this correct or there is gap in my thesis?

A:

Take :
$$\tag{1}(x,y,z) \in H \to (\tfrac{x}{\sqrt{x^2+y^2}},\tfrac{y}{\sqrt{x^2+y^2}},z) \in C$$
and its reciprocal transformation:
$$\tag{2}(u,v,z) \in C \to (u\sqrt{1+z^2},v\sqrt{1+z^2},z) \in H$$
Explanation : the intersection of plane $z=z_0$ with :

cylinder $C$ is a unit radius circle.
one-sheeted hyperboloid $H$ is a circle with radius $\sqrt{1+z_0^2}$ (do you see why ?).

Then it remains to map radially one circle onto the other, which is what  transformations (1) and (2) are doing.

