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.

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 $$0$$.
• But your answer says $-108$ is excluded? I think you intended as in (5), Prof Tu? – Selene Auckland May 28 at 23:35