Timeline for When can we factor out the time dimension?
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 15, 2010 at 23:09 | vote | accept | Daniel Litt | ||
| Jul 15, 2010 at 23:09 | comment | added | Daniel Litt | Indeed, and this can certainly be done smoothly in t. +1, accepted. | |
| Jul 15, 2010 at 22:42 | comment | added | user1504 | Take a path connecting the first point to the second point. Thicken the path to a neighborhood. Said neighborhood is diffeomorphic to the ball. It should at least be intuitive that you can always map the ball onto itself in a way that moves any given point to the origin. Just grab the point and drag it where you want it. | |
| Jul 15, 2010 at 22:40 | comment | added | Daniel Litt | Should this claim be obvious? It certainly suffices. (This is where my lack of diff. geom. knowledge is apparent.) | |
| Jul 15, 2010 at 22:30 | comment | added | user1504 | I still suspect the two conditions are essentially equivalent, as long as you're not worried about metric structure. We know that $M$ is globally hyperbolic, so let's equate $M = S \times \mathbb{R}$. Let $f_0$ be the "base" path, given by $t \mapsto (f_0(t),t)$. Now suppose we have some other path $f_1$, given by $t \mapsto (f_1(t),t)$. I claim that, as long as $S$ is path-connected, for each $t$, we can find a diffeomorphism $d_t: S \to S$ which moves $f_1(t)$ to $f_0(t)$. Moreover, we can make these diffs depend smoothly on $t$. That should give product structure you want... | |
| Jul 15, 2010 at 20:57 | comment | added | Daniel Litt | I'm worried that the projection of $f$ to $S$ might not be constant. If you read the question carefully, you'll see we need a different product structure for each $f$. | |
| Jul 15, 2010 at 20:54 | comment | added | user1504 | You're worried that $S$ may depend on $f$? It might in the metric category, but in the topological category, any two $S$ are certainly isomorphic. | |
| Jul 15, 2010 at 20:04 | comment | added | Daniel Litt | Reading the wikipedia article, it seems to me that this implies that one path can be factored out in the sense I described; does this guarantee it for every path? | |
| Jul 15, 2010 at 20:01 | history | answered | user1504 | CC BY-SA 2.5 |