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Does there exist a submersion $f:\mathbb{R}^{3}\setminus\{0\} \to \mathbb{R}$ for which there are $c_1$ and $c_2$ in $\mathbb{R}$ such that $f^{-1}(c_1)$ is compact and $f^{-1}(c_2)$ is non-compact.

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    I can imagine one when you look for submersions $f:\mathbb{R}^3\setminus\lbrace -1,+1\rbrace\rightarrow\mathbb{R}$, but not when you only take out one point... I think the answer may be no.2011-07-25
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    One could try to look for a function $\mathbb{R}^3\setminus\lbrace0\rbrace\to\mathbb{R}^*_+$ whose level sets would look like spheres for small $0$c\to 1$, so that they "hug" an infinite vertical cylinder, be that infinite vertical cylinder for $c=1$, and be vertical cylinders of increasing radii as $c\to\infty.$ So it might be possible after all... – 2011-07-25
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    Something like $f(x,y,z)=(x^2+y^2)$ when $x^2+y^2\geq 1$, and $\frac{x^2+y^2+z^2(1+z^2)\varphi(x^2+y^2))}{x^2+y^2+z^2(1+z^2)}$ where $\varphi:\mathbb{R}\to\mathbb{R}$ is a smooth function that is increasing, $\geq 0$, constantly $=0$ on some neighborhood of $0$ and $\varphi(x)=1+\lambda\times (1-x)$ on some neighborhood of $1$. One would need to do some refinements, maybe add some constants here and there and modify $f$ and $\varphi$ to make $f$ globally smooth, but this should essentially work.2011-07-25
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    Oh no xD, the formula I gave doesn't do the job... I'll think about it :D.2011-07-25

2 Answers 2

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Why not try something simpler, like $f=\frac{x^{2}+y^{2}+z^{2}}{1+z^{2}}$. Then it is easy to see that $\operatorname{grad}f$ is a non-zero vector at each point of $\mathbb{R}^{3}\backslash\{0\}$, so $f$ is a submersion; on the other hand

the level set $f^{-1}(1):~x^{2}+y^{2}=1$ is a cylinder, hence noncompact, while

the level set $f^{-1}(\frac{1}{2}):~x^{2}+y^{2}+\frac{1}{2}z^{2}=\frac{1}{2}$ is an ellipsoid, hence compact.

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This is a simpler example than what I had.

Let $U=\{(x,y)\in\mathbb R^2:0

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    It seems to me this is not an example. As I see it, you are foliating a half space missing a closed half line, which is diffeomorphic to a full space, not a space missing a point. More importantly, all leaves are non compact. Am I misunderstanding your example?2011-07-25
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    @Olivier: I changed the example for something that more obviously works.2011-07-25
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    Do you know if every codimension one foliation that also verifies that every point has a "good" neighborhood (that is, any single leaf intersects it in a single path connected component, unlike what happens on a torus foliated by an irrational slope, where intersecting one leaf with a neighborhood yields infinitely many path components) can be obtained via a submersion $M\to\mathbb{R}$?2011-07-25
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    @Olivier: consider the Reeb foliation of the solid torus, as pictured on http://en.wikipedia.org/wiki/File:Reebfoliation-ring-2d-2.svg This can be lifted to a foliation of a cilinder $\mathrm{Disk}\times\mathbb R$, which can then be extended to a foliation of the whole of $\mathbb R^3$ by foliating the exterior of the solid cylinder with cilinders. The resulting foliation of $\mathbb R^3$ satisfies your condition, but the leaves are not level surfaces of a function.2011-07-25
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    (Because all leaves accumulate on the boundary of $\mathrm{Disk}\times\mathbb R$, so any candidate function would have to be constant on the solid cylinder)2011-07-25
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    OK, I agree! One could directly go for $U=\lbrace (x,y,z)\in\mathbb{R}^3| 0$U$ with the punctured $3$ space. This way, you don't need to rotate. – 2011-07-25
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    Yeah, that works :)2011-07-25
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    I agree with you that the Reeb foliation of the solid torus cannot be obtained as the level sets of a submersion, but I don't understand why the foliation of $3$ space you describe cannot be. For the solid torus, a single leave revolves infinitely often and passes infinitely often by a *fixed* point in the boundary, and continuity would imply that the submersion would have to be constant on the solid torus, OK. But this argument fails with the foliation you describe. I can see that the further down you travel along the limit cylinder, the steeper the gradient must be...2011-07-25