Timeline for Boundary of fibers of submersions
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Mar 5, 2014 at 14:24 | history | undeleted | Hugo Chapdelaine | ||
| Mar 5, 2014 at 14:24 | history | deleted | Hugo Chapdelaine | via Vote | |
| Mar 5, 2014 at 14:23 | comment | added | Hugo Chapdelaine | So I thought about the example about the solid cylinder in $R^3$ (of length $1$) along the $x$-axis and center at $(0,0,0)$ where $f$ is the distance between the projection of a point $P$ on $x$ and $(0,0,0)$. But I found this example a bit artificial since in this case, $N$ is really just $[-1/2,1/2]$ since the fibers outside this interval are empty. But now I just realize that one may identify the two end points of that interval to get a circle. Ok, I see so anything can happen. I guess this answers my question. | |
| Mar 5, 2014 at 14:10 | history | edited | Hugo Chapdelaine | CC BY-SA 3.0 |
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| Mar 5, 2014 at 13:49 | history | edited | Hugo Chapdelaine | CC BY-SA 3.0 |
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| Mar 5, 2014 at 10:58 | comment | added | user126154 | if you remove the hypothesis that f is regular on the boudary, then $X=f^{-1}(x)\cap M$ is a, not necessarily proper, sub-manifold of the interior of $M$. In general $\partial X$ may empty, defined, or even not being well-defined in the sense of manifolds. If $X$ accumulates to the boundary, the accomulation points belong to $f^{-1}(x)\cap\partial M$, but can be smaller than that (example: take a cylynder, collapse the boundary components to points to get a spehre and map it to R) | |
| Mar 5, 2014 at 6:20 | comment | added | Ryan Budney | Take a compact disc, and any real-valued function on it with no critical points. That's a pretty typical example. | |
| Mar 5, 2014 at 2:08 | history | edited | Hugo Chapdelaine |
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| Mar 4, 2014 at 19:12 | history | asked | Hugo Chapdelaine | CC BY-SA 3.0 |