Consider the set $S \subset \mathbb{R}^4$, define by the equations, $x^2 + y^2 = 1$ and $ z^2 + t^2 = 1$.
a) Prove that $S$ is a regular surface.
b) Find a parameterization of $S$ around the point $p = (1, 0, 1, 0)$ and calculate the 1st fundamental form.
(Im new with differential geometry and I do not really know how to start this exercise because is $\mathbb{R}^4$, many thanks!)
Consider $f:\mathbb{R}^4\rightarrow\mathbb{R}^2$ defined by $f(x,y,z,t)=(x^2+y^2,z^2+t^2)$. Show that it is a submersion in a neighborhood of $f^{-1}(1,1)$ by computing the Jacobian. This implies that $f^{-1}(1,1)$ is a submanifold.