I was trying to prove the following:
Let $F(x,y)=x^3+xy+y^3$ be defined on $\mathbb R^2$. Show that $F^{-1}[\{0\}]$ is not an embedded submanifold of $\mathbb R^2$.
I have verified that $(0,0)$ is not a regular point of $F$ and it is a saddle point. I have read in another question that this implies $F^{-1}[\{0\}]$ is the image of a smooth curve with self-intersections.
I would like to solve this problem without using this "result", since I don't know a proof of it and haven't found one anywhere, or else I'd like a reference to such a proof.
Best Answer
As you noticed, $p:=(0,0)$ is a non-degenerated critical point of $F$, which means that $\nabla_pF=0$ and $H_pF$, the hessian matrix of $F$ at $p$ is invertible, indeed, one has the following equality: $$H_pF=\begin{pmatrix}0&1\\1&0\end{pmatrix}.$$ Therefore, using Morse's lemma (Taylor's formula+inverse function theorem), up to a change of coordinates sending the origin on the origin, in a neighborhood of $(0,0)$, one has that: $$F(x,y)=x^2-y^2.$$ From there it is clear that $S:=F^{-1}(\{0\})$ is not a manifold. For example, you can notice that $S\setminus\{(0,0)\}$ has $4$ connected-components, exactly as the set $\{x=y\}\cup\{x=-y\}$ minus the origin.
Here is a sketch of $S$:
As you can see, in a neighborhood of $(0,0)$, $S$ can be straightened onto $\{x=y\}\cup\{x=-y\}$, which is exactly what we have done.