Is $x^3-6xy+y^2=-108$ a regular submanifold but not a regular $k$-submanifold?

My book is An Introduction to Manifolds by Loring W. Tu. Let $S = \{x^3-6xy+y^2=-108\}$, and let "submanifold" and "$k$-submanifold" mean, respectively, "regular" and "regular $k$-submanifold".

As in here, we have that Tu's manifolds with or without boundaries do not necessarily have dimensions. Do Tu's (regular) submanifolds, however, necessarily have dimensions?

• Here is Definition 9.1 of (regular) submanifolds, which seems to have dimensions.

• But now consider Problem 9.1: The answer given is all real numbers except $0$ and $-108$. A solution given by Richard G. Ligo claims that the reason (or a reason) why $x^3-6xy+y^2=-108$ is not a (regular) submanifold of $\mathbb R^2$ is that connected components do not have the same dimension.

I think we must have either that

1. Ligo's solution is wrong.

2. Tu's submanifolds have dimensions and so $S$ is not a submanifold (i.e. $k$-submanifold, in this case) of $\mathbb R^2$ because of the connected components and no other reason.

3. Tu's regular submanifolds have dimensions and so $S$ is not a submanifold of $\mathbb R^2$ because of the connected components, but there are other reasons why $S$ is not a $k$-submanifold of $\mathbb R^2$.

4. Tu intended a definition that allows submanifolds to not have dimensions. However, $S$ is neither a submanifold nor a $k$-submanifold of $\mathbb R^2$ for a different reason.

5. Tu intended a definition that allows submanifolds to not have dimensions and should have allowed $S$ to be a submanifold of $\mathbb R^2$ even though $S$ is not a $k$-submanifold of $\mathbb R^2$. Thus each nonzero $c$ gives a submanifold with or without uniform dimension (same dimension for each connected component), while $c=-108$ is the only nonzero value that gives submanifold without uniform dimension.

Update: I asked

Hello Prof Tu, I replied on stackex, but anyway for your convenience: It's actually just that your answer excluded -108. I think you meant to exclude -108 for submanifold with uniform dimension (same dimension for each connected component) but to include -108 for submanifold with non-uniform dimension? ...,

and Prof Tu replied

...The critical values are 0 and -108, but the inverse image of -108 is a regular submanifold. Your interpretation is correct...

Solution 1:

To me, a manifold or a submanifold can have connected components of different dimensions, so the set in question is a regular submanifold of $\mathbb{R}^2$ with one connected component of dimension $1$ and one connected component of dimension $0$.