For Which Additional constant a Manifold is guaranteed?

Question

Set is given by $M = (x,y) \in \mathbb{R}^2 \vert \underbrace{e^{x^2+2\,y^2+2}}_{f} = c$.

Instantly I noticed $M$ can be counted as Manifold if $c \in(e^2,\infty)$ because $f \geq e^2$. Now apparently this is not complete, so I wonder which additional c I have to take into consideration. Is it the boundary $[e,\infty)$ ? Or even other sets.

score 1 · Accepted Answer · answered Feb 13 '21 at 19:41

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We have $M=\{(x,y)\in \mathbb{R}^{2} \colon x^{2}+2y^{2}=d\}$, where $d=\operatorname{log}(c)-2$. Let $g=x^{2}+2y^{2}$. We have that $\operatorname{grad}(g)=(2x,4y)$ vanishes at $(0,0)$. Now let's look at some cases:

If $d<0$ (which is the same as $c<e^{2}$), $M=\emptyset$ isn't a manifold.

If $d=0$ (i.e. $c=e^{2}$) then $M=\{(0,0)\}$ is a point, which is a manifold as long as you admit zero-dimensional manifolds.

If $d>0$ (which is the same as $c>e^{2}$), then $\operatorname{grad}(g)(x,y)\neq 0$ for every $(x,y)\in g^{-1}(c)$, so by the regular level set theorem, $g^{-1}(c)$ is a ($1$-dimensional) embedded submanifold of $\mathbb{R}^{2}$.

Hope this helps!

answered Feb 13 '21 at 19:41

Johnny El Curvas

1,592

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Why do you think the empty set is not a manifold? – Moishe Kohan Feb 13 '21 at 19:43
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For me, manifolds are supposed to be nonempty, but you could also argue that $\emptyset$ is (vacuously) an $n$-manifold for every $n\geq 0$. – Johnny El Curvas Feb 13 '21 at 19:44
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Yes, the empty set us a manifold of any dimension. – Moishe Kohan Feb 13 '21 at 19:54
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By that logic, then $M$ is always a manifold. Now it would be a matter of "what do we understand to be a manifold", but I think what was expected of this question is to exploit the Regular Level Set Theorem. – Johnny El Curvas Feb 13 '21 at 20:00
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So this confirmed my assumption. I gotta include the empty and pointy set as manifold. – Leon Feb 13 '21 at 20:17
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Do I have to argue it's a manifold for each value c accordingly ? – Leon Feb 13 '21 at 20:20
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or $c \in \mathbb{R} \setminus e^2 $ ? – Leon Feb 13 '21 at 20:24
I'd say for each value, since they have different dimensions and are manifolds for "different reasons". – Johnny El Curvas Feb 13 '21 at 20:59

For Which Additional constant a Manifold is guaranteed?

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