So, I would like to prove that the curve $\alpha :y^2 -x^3 =0$ is not a differential submanifold of $\mathbb{R}^2$.
My notes are quite messy about this, and at the time it was an argument that I really didn't get. Moreover, it's one of the first times I have to deal with manifolds. I know that I am supposed to use (I mean, the teacher used) the implicit function theorem, and see the curve as the locus of zeros of a differentiable function in $\mathbb{R}^2$, because I need a submanifold of $\mathbb{R}^2$. If you consider such a function, you can prove that in $(0,0)$ both partial derivatives are zero. Then you say you can't apply the implicit function theorem and so the curve is not a submanifold of $\mathbb{R}^2$.
There are a few things I am not sure about. However, the most troublesome is by far the application of the implicit function theorem. I mean, the theorem is great if you want to prove that some curve has a regular parametrization without bothering searching for an explicit one, which is great if I had wanted to prove that a curve is a differential submanifold. Here I cannot apply the theorem in $(0,0)$. How can you conclude then? Doesn't the implicit function theorem give only sufficient conditions? I mean, if you prove that every differentiable function in $\mathbb{R}^2$ with $\alpha$ as locus of zeros has partial derivatives equal to $0$ at the origin, why should it mean that no structure of differential submanifold is possible at all?
I have made @TedShifrin's argument rigorous here. Suppose for the sake of contradiction that the curve $y^{2}-x^{3} = 0$ is a smooth manifold. In this case we may choose $\eta\in(0, 1)$ such that $\Gamma:=\{\|x\|_{\infty} < \eta\}\cap\{y^{2}-x^{3} = 0\}$ is diffeomorphic to an open interval $I$ (since this set is connected). Let $g = (g_{1}, g_{2}): I\rightarrow\Gamma$ denote this diffeomorphism. Note that the function $g_{2}: I\rightarrow(-\eta^{3/2}, \eta^{3/2})$ is smooth, injective, and in particular, $g_{2}'\neq 0$ on $I$. Therefore, using the Inverse Function Theorem, it is a diffeomorphism and thus $g\circ g_{2}^{-1}:(-\eta^{3/2}, \eta^{3/2})\rightarrow\Gamma$ is a diffeomorhism. For any $t\in(-\eta^{3/2}, \eta^{3/2})$ we have $g\circ g_{2}^{-1}(t) = (g_{1}(g_{2}^{-1}(t)), t)\in\Gamma$ and thus we must have $g_{1}(g_{2}^{-1}(t)) = t^{2/3}$. This implies that $g\circ g_{2}^{-1}(t) = (t^{2/3}, t)$ is a smooth, which is clearly false.