Let $f:M \to N$ be a surjective submersion. I want to show that that the graph defined as $$G = \{(x,y) \in M\times M | f(x) = f(y)\}$$ Is an embedded submanifold of $M \times M$. I have some ideas but I haven't been able to complete the proof.
Let $m,n$ be the dimensions of $M,N$ respectively. Then I was thinking to show that for there exists a $k$ such that for every point in $G$ there exists a chart $(U, \phi)$ such that $\phi(U \cap G) = \{(x^1,....,x^{2m})| x^{k+1} =...= x^{2m} = 0\}$
Now I know that if $(p_1,p_2) \in G$ then by the constant rank thereom I know that there exists charts $(U_1,\phi_1), (U_2,\phi_2) $ for $p_1,p_2$ respectively and chart $(V,\psi)$ on $N$ for $f(p_1)=f(p_2)$ such that $$\psi \circ f \circ \phi_1^{-1}(x^1,..,x^n,..,x^m) = (x^1,...,x^n)$$ $$\psi \circ f \circ \phi_1^{-1}(y^1,..,y^n,..,y^m) = (y^1,...,y^n)$$
** I only changed the variable because we could be working in different charts
Now from here, I was thinking to try to use the proof implicit function theorem to try and create slice charts $f^{-1}(p_1) \cap U_1$ and $f^{-1}(p_1) \cap U_2$. Then applying $\phi_1, \phi_2$ respectively on them to get that they map (from local coordinates) $(x^1,....,x^m) \to (0,0...,x^{n+1},...,x^m)$ and $(y^1,...y^m) \to (0,..,0, y^{n+1},...,y^m)$ then I could use the propduct of those two charts and slice charts to create slice charts for $G$. However In the proof that I've seen for the implicit function theorem, we assumed without loss of generality that if $q = f(p_1)$, and $(V, \psi)$ is a chart with $q \in V$, then $\psi(q) = 0$. But I don't think I can do this in this case, because we cannot assume that for every value $q \in N$. And also this would give me $0$ in the first $n$ coordinates, which is slightly different than our definition of embedded submanifold that needed the last coordinates to be $0$. Or perhaps where the $0$ are in the coordinates do not matter as much as I think. But even then, I'm not sure if this is the way to go about it. I'm new to differential geometry and I'm trying to piece this all together.
My second idea was to create a map $H(x,y) = F(x)-F(y)$ that maps $M \times M \to N$, and then use the implicit function theorem on the point $0$, because $H^{-1}(0)$ is exactly our graph $G$. But I'm not even sure if $0$ is an element of $N$ or if I can even prove such a map is a submersion.
Lastly, I was thinking to create a map $G \to M \times M$ that is an embedding but I haven't been able to think of one, or think about how to even prove such a map is an embedding.
Any help would be fantastic..