Timeline for I was wondering if the set of singular loops is a (somewhere) submanifold of loop space?
Current License: CC BY-SA 3.0
23 events
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| Sep 18, 2011 at 18:55 | vote | accept | Nikita Kalinin | ||
| Sep 18, 2011 at 18:55 | history | bounty ended | Nikita Kalinin | ||
| Sep 18, 2011 at 18:55 | comment | added | Nikita Kalinin | to Ryan: I was testing this example before, and it was incorrect :) So, it was the reason of my question. Thank you very much, you have completely answered to all my questions. | |
| Sep 18, 2011 at 18:03 | comment | added | Ryan Budney | Well, the "uniqueness" of the local minimum is that it's that $d(g(t_1),g(t_2))$ is the smallest among all local minima. | |
| Sep 18, 2011 at 17:47 | comment | added | Ryan Budney | Consider $f \in D(M) \setminus D'(M)$ as in my answer. Let $U$ be a neighbourhood of $f$ in $L(M)$ and $g \in U$. You can argue if $U$ is small enough, the function $d(g(t_1),g(t_2))$ where $d : M^2 \to \mathbb R$ is the metric on $M$ and $t_1 \neq t_2 \in S^1$ has a unique local minimum. So if $U$ is small, using charts you can consider $g(t_1)-g(t_2)$ to be a vector, and project it into the orthogonal complement of $img(Df_p)\oplus img(Df_q) \subset T_pM$. This is the map you're looking for. | |
| Sep 18, 2011 at 17:41 | comment | added | Ryan Budney | @Nikita: in the situation I describe in my answer, the map you ask for exists. There's a general-nonsense construction of the map given by inverting the chart I describe above and projecting onto the factor corresponding to the bump function perturbation. But there's also a direct construction from differential geometry. (see next comment) | |
| Sep 18, 2011 at 13:36 | comment | added | Nikita Kalinin | to Ryan: Thank you very much! so, I have the last question about local "submanifoldability" of a $D\setminus D'$. For a smooth submanifold $X\subset Y$ of codimension 2, for a general point $x\in X$ we always have a map from small neghbourhood of $x$ to $D^2$ ($x\in U\subset Y, f:U\to D^2$) such that $U\cap X = f^{-1}(0)$. I belive that in situation of space of loops there is no such map... So, you have described local charts of $D\setminus D'$ but whether in this situation exist map $f$ described above? | |
| Sep 16, 2011 at 20:55 | comment | added | Andrew Stacey | Ryan, I wrote up some stuff on this on the nLab: ncatlab.org/nlab/show/… I'd be interested in your thoughts. | |
| Sep 16, 2011 at 4:08 | comment | added | David Roberts♦ | Try \ast instead of * | |
| Sep 15, 2011 at 19:11 | history | edited | Ryan Budney | CC BY-SA 3.0 |
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| Sep 15, 2011 at 18:01 | history | edited | Ryan Budney | CC BY-SA 3.0 |
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| Sep 15, 2011 at 17:55 | history | edited | Ryan Budney | CC BY-SA 3.0 |
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| Sep 15, 2011 at 17:54 | comment | added | Ryan Budney | I'm describing a chart in the Frechet context. | |
| Sep 15, 2011 at 17:49 | history | edited | Ryan Budney | CC BY-SA 3.0 |
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| Sep 15, 2011 at 17:44 | comment | added | Nikita Kalinin | ok. It seems that you understand my vague question! So, $D\setminus D'$ is a submanifold... In which sense?(any graph of function $\mathbb R \to\mathbb R$ is a "submanifold" of a plane: graph has a "normal bundle"...) Is it submanifold in sense of Fréchet or Frolicher? Is it diffeological submanifold of loop space? | |
| Sep 15, 2011 at 17:24 | history | edited | Ryan Budney | CC BY-SA 3.0 |
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| Sep 15, 2011 at 17:06 | comment | added | Ryan Budney | @Nikita, that's what the first half of my response addresses. Yes, if $f$ has only a regular double point, there is submanifold chart for $D(M)$ in a neighbourhood of $f$. The chart neighbourhood (that "flattens" $D(M)$) is given by this deformation. | |
| Sep 15, 2011 at 15:20 | comment | added | Nikita Kalinin | to Ryan: see EDIT2 in main post, please. | |
| Sep 15, 2011 at 15:15 | comment | added | Nikita Kalinin | I feel that I explain very bad :( Sure, there are examples where $D$ near a point is not a submanifolds. Question is about opposite situation. For example, let's consider loop $f$ which has branch (-t,t,0) and branch (t,t,0) and have no other singularities. Is it true that $D$ near the $f$ is a submanifold? Yes, sure, there is a one-parameter versal deformation of $f$ and any deformation of $f$ induced from that one. Does it imply that $D$ near the $f$ is a submanifold? | |
| Sep 14, 2011 at 23:02 | comment | added | Ryan Budney | I've made my answer a little more explicit. It is describing a "local situation". | |
| Sep 14, 2011 at 23:01 | history | edited | Ryan Budney | CC BY-SA 3.0 |
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| Sep 14, 2011 at 18:24 | comment | added | Nikita Kalinin | yes, you are right, but my question is about local situation. Does only one point $f_0\in D$ exist such near the $f_0$ $D$ looks like submanifold? | |
| Sep 14, 2011 at 17:57 | history | answered | Ryan Budney | CC BY-SA 3.0 |