Finding a smooth atlas of $M=\{ (x,y,z) \in \mathbb{R}^3 \; | \; x^2+y^2-z^2=1 \}$ We know that $M=\{ (x,y,z) \in \mathbb{R}^3 \; | \; x^2+y^2-z^2=1 \}$ is a submanifold of $\mathbb{R}^3$, with dimension $2$. The point is that I have to find an atlas which define the structure of $C^{\infty}$ manifold for $M$. I just discovered these objects, and I have to say that I don't know how to do. As $M$ is a submanifold of dimension $2$, I think I should find some open sets $(U_i)_i$ of $M$, and some open sets $(V_i)_i$ of $\mathbb{R}^2$, and some homeomorphism $\psi : U_i \rightarrow V_i$, such that the transition map are $C^\infty$.
My first idea was to use the theorem of the implicit functions, to have $z$ function of $x,y$ and then I would be able to solve the problem I think. But all I know is that : $(x,y,z) \in M \iff z = +- \sqrt{x^2+y^2-1}$, and from that, I tried several things but it doesn't lead where I wanted.
Someone could help me, and explain me how to do, and what's the idea between thoses objects ?
Thank you !

The space is topologically equivalent to a punctured plane, which suggests that we can cover $M$ with a single smooth chart. Since the defining equation has the form $F(x^2 + y^2, z) = C$, $M$ is invariant under rotations about the $z$-axis $Z$, which suggests that we look for a map $M \to \Bbb R^2 - \{ 0 \}$ that is suitably compatible with those rotations (or more precisely, that is equivariant for the two rotation actions.)
In particular, this suggests that if we view $\Bbb R^2$ as the $xy$-plane in $\Bbb R^3$ we map the arc given by the intersection of $M$ with a half-plane bounded by $Z$ to the ray $\Bbb R^2 \cap Z$ of the form
$$(x, y, z) \mapsto \left(f(z) \frac{x}{|x|}, f(z) \frac{y}{|y|}\right)$$
for a suitable bijection $f : \Bbb R$ to $\Bbb R^+$. A natural choice is $f(z) := \exp z$, giving
$$(x, y, z) \mapsto \left(\frac{x}{|x|} \exp z, \frac{y}{|y|} \exp z\right) .$$
