Find an explicit atlas for this submanifold of $\mathbb{R}^4$

Question

I'm having a hard time coming up with atlases for manifolds. I can prove using the implicit function theorem that

$M = \{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4:x_1^2+x_2^2=x_3^2+x_4^2=1\}$

is a $2$-dimensional manifold. I would like to find an explicit atlas for this manifold now. Any help would be greatly appreciated.

Answer 1

OK, so based on the comments, I think this should be the answer:

Let

$U_1=\{(x_1,x_2,x_3,x_4)\in M: x_1>0\}, \phi_1:U_1\to\mathbb{R}$ defined as $\phi(x_1,x_2,x_3,x_4)=x_2$,

$U_2=\{(x_1,x_2,x_3,x_4)\in M: x_2>0\}, \phi_2:U_2\to\mathbb{R}$ defined as $\phi(x_1,x_2,x_3,x_4)=x_1$,

$U_3=\{(x_1,x_2,x_3,x_4)\in M: x_1<0\}, \phi_3:U_3\to\mathbb{R}$ defined as $\phi(x_1,x_2,x_3,x_4)=x_2$,

$U_4=\{(x_1,x_2,x_3,x_4)\in M: x_2<0\}, \phi_4:U_4\to\mathbb{R}$ defined as $\phi(x_1,x_2,x_3,x_4)=x_1$.

Then, we'll do the same thing with the third and the fourth components (calling them $V_j$ and $\psi_j$) and an atlas for $M$ should consist of $\{(U_i\times V_j,\phi_i\times \psi_j)\}_{i,j=1}^4$.

Is this the correct answer?