Prove that a manifold given by a system of equations in $\mathbb{R}^4$ is diffeomorphic to the 2-torus $T^2$

Question

We have a system of equation in $\mathbb{R}^4$ given by $$\begin{cases} &x^2+y^2 +u^2+v^2 =1 \\ &x^2 + y^2 - u^2 - v^2 =0. \end{cases}$$ We were first asked to prove that this is a smooth manifold which I did by finding the Jacobian matrix:

$$J = \begin{pmatrix} 2x & 2y&2u&2v \\ 2x&2y&-2u&-2v \end{pmatrix}$$

then I found the matrix $JJ^T$ and shown that this matrix wasn't singular since $x,y,u,v$ could never be such that the determinant of $JJ^T$ was 0 (please correct me here if this isn't correct.). The dimension of the manifold I then found to be 2.

I was then asked to show that this manifold is diffeomorphic to the 2-torus and I have no idea how to do this.

Answer 1

You can see the torus $T^2$ as the subset of $\mathbb{R}^4$ defined by $(x,y,u,v), x^2+y^2=1, u^2+v^2=1$.

Define $f(x,y,u,v)=\sqrt2(x,y,u,v)$ and $g(x,y,u,v)={1\over\sqrt2}(x,y,u,v)$, defines a diffeomorphism and its inverse between the manifold and the torus.