TITLE: Structure of a $ C^{\infty} $-manifold
QUESTION [8 upvotes]: I was studying differentiable manifolds (an introduction) and found the following example, but I am confused.
Example
The function
\begin{align}
f: &\mathbb{R}^{3} \to \mathbb{R}, \\
f: &(x,y,z) \mapsto x^{3} + 2 y^{3} + z^{3} + 6 x^{2} y - 1
\end{align}
defines the structure of a $ C^{\infty} $-manifold on $ {f^{-1}}(0) $.
Thank you.

REPLY [3 votes]: Guillemin and Pollack would go about this by defining a regular value: A regular value of a map $f:X\to Y$ is any point $y\in Y$ such that the derivative of $f$ is nonsingular at points mapping to $y$. Then they show that the preimage $f^{-1}(y)$ is a submanifold of $X$. (Differential Topology, p. 21)
So to solve your question this way, you just need to compute the Jacobian of $f$ and find when it is singular, then you show that none of these points are in $f^{-1}(0)$. Then you know $0$ is a regular value and $f^{-1}(0)$ is a manifold.
Note that the regular value method subsumes the machinery used in the other answer (i.e. arguments like that are used to prove the things leading up to the regular value theorem.)