Is $M=\{(x,y,z)\in \mathbb{R}^3: x^3+y^3+z^3=1, z=xy\}$ a manifold of class $C^{\infty}$?

Question

Let $M=\{(x,y,z)\in \mathbb{R}^3: x^3+y^3+z^3=1, z=xy\}$. Is $M$ a manifold of class $C^{\infty}$?

I need find a atlas $\{(U_i,\varphi_i)\}_{i\in I}$ with $U_i$ open sets and $\varphi$ diffeentiable, but I not can find this atlas.

Thanks for your help.

Answer 1

Hint: Show that $f:R^3\rightarrow R^2$ defined by $f(x,y,z)=(x^3+y^3+z^3-1,z-xy)$ is a submersion.

$df=\pmatrix{ 3x^2 & -y\cr 3y^2 & -x\cr 3z^2 &1}$.

It is not a submersion implies by computing the minors $x^3+y^3=0, x^2+yz^2=0, y^2+xz^2=0$. This implies that by multipliying the two last equations respectively by $x$ and $y$ $x^3=-xyz^2, y^3=-xyz^2$, But the first equation says $x^3=-y^3$, we deduce $x^3=y^3=0$, so $x=y=z=0$. Thus $f$ is a submersion on $R^3-\{(0,0,0)\}$ and $f^{-1}(0,0)$ is a submanifold since it does not contains $(0,0,0)$.