Submanifold of $S^1 \times S^1 \times \mathbb{R}^2$.

Question

Let's $f,g : S^1 \longrightarrow \mathbb{R}^2$ be embeding's. Consider $F: S^1 \times S^1 \times \mathbb{R}^2 \longrightarrow \mathbb{R}^2 $, $F(x,y,v)= f(x) - g(y) -v$. Show that $N= F^{-1}(0)$ is submanifold of $S^1 \times S^1 \times \mathbb{R}^2$.

Answer 1

You need only check that $F$ has rank $2$ at every point $(x,y,v) \in S^1 \times S^1 \times \mathbb{R}^2$, i.e. $dF$ on tangent spaces is a surjection everywhere it's defined. In fact, one only needs to check that $dF$ is a surjection on $F^{-1}(0)$, but we'll prove the stronger version, so that $F^{-1}(r)$ is an embedded submanifold of $S^1 \times S^1 \times \mathbb{R}^2$ for all $r \in \mathbb{R}^2$.

Consider the map $h: \mathbb{R}^2 \to S^1 \times S^1 \times \mathbb{R}^2$ given by $h(t) = (x,y,t)$. Then $F \circ h(t) = f(x) - g(y) -t$, so $F \circ h$ is a diffeomorphism from $\mathbb{R}^2$ to itself. Therefore the tangent map $d(F \circ h)$ is an isomorphism, but $d(F \circ h) = dF \circ dh$, so that $dF$ is in fact a surjection $T_{(x,y,v)}(S^1 \times S^1 \times \mathbb{R}^2) \to T_{-v}(\mathbb{R}^2)$.