TITLE: Embedded submanifold of $\mathbb{S}^3$
QUESTION [6 upvotes]: Let's consider $f:\mathbb{S}^3\to \mathbb{R}$ with $f(x,y,z,t)=x^2+y^2-z^2-t^2$
(1) show that $M_0:= f^{-1}(0)$ is an embedded submanifold of $\mathbb{S}^3$.
(2) show that $M_0$ is diffeomorphic to the 2-torus.
(3) find all points at which $f$ is a submersion.
I have some problem with this exercise because I want to use the regular value theorem but proving that $f$ is a submersion at all points of $f^{-1}(0)$, with charts of $\mathbb{S}^3$, needs way too many calculations. How can I solve smartly this problem?
My attempt for points (1) and (3):
I can consider the natural extension of $f$ to $\mathbb{R}^4$ given by $\widetilde{f}: \mathbb{R}^4\to \mathbb{R}$ with $\widetilde{f}(x,y,z,t)=x^2+y^2-z^2-t^2$. Clearly $\widetilde{f}$ is a submersion at all points $p\neq 0$, so also at every $p\in \mathbb{S}^3$. Therefore, since $f=\widetilde{f}\circ \iota\mid_{\mathbb{S}^3}$, by chain rule I get, for $p\in \mathbb{S}^3$,
$$(df)_p=(d\widetilde{f})_p\circ (d\iota\mid_{\mathbb{S}^3})_p$$
where I know that $(d\widetilde{f})_p$ is surjective and $(d\iota\mid_{\mathbb{S}^3})_p$ is injective. It seems to be almost done but I don't know how to conclude.

REPLY [2 votes]: Hint:Let $G:\mathbb{R}^3\to\mathbb{R}$ denote the function $G(x,y,z,t)=x^2+y^2+z^2+t^2-1$.  We have the following:

$T_p\mathbb{S}^3=\ker dG_p$ for $p\in\mathbb{S}^3$.
$p\in\mathbb{S}^3$ is a critical point of $f$ if and only if $T_p\mathbb{S}^3\subset\ker d\tilde{f}_p$. By dimensional consideration, this is true if and only if $T_p\mathbb{S}^3=\ker d\tilde{f}_p$.
Two linear functionals on a vector space are constant multiples of each other if they have the same kernel.