Is the set $\{ x,y \in \mathbb{R} \,|\,x^3+y^3 = 1\}$ a manifold?

Question

I think $\{ x,y \in \mathbb{R} \,|\,x^3+y^3 = 1\}$ is a manifold in $\mathbb{R}$, specifically using the function $f:\mathbb{R^2}\to\mathbb{R}$ defined as $f(x,y)=x^3+y^3 -1$. It has the property that the differential has full rank everywhere except $(0,0)$, which isn't included in the set and the intersection with the set and $\mathbb{R^2}$ is exactly when $f(x,y) = 0$. Is this correct?

Answer 1

It is correct. Here is a theorem you need (cited from Lee's Introduction to Smooth Manifolds):

Theorem 5.12 (Constant-Rank Level Set Theorem) Let $M$ and $N$ be smooth manifolds, and let $\mathcal{\Phi}: M \rightarrow N$ be a smooth map with constant rank $r$. Each level set of $\Phi$ is a properly embedded submanifold of codimension $r$ in $M$.

In your case, $M= \mathbb{R}^2\setminus\{0\}$, $N = \mathbb{R}$, $r = 1$.