Timeline for Synthetic differential / conformal geometry of Lorentzian manifolds?
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| May 12, 2023 at 1:28 | comment | added | RBega2 | Yes. A good example is convexity in one variable analysis. By this I mean, $f(tx+(1-t) y)\leq tf(x)+(1-t)f(y)$ for $t\in [0,1]$. When the function is $C^2$ this is equivalent to $f''\geq 0$, but the first notion makes sense even for less smooth things (and while it does impose some regularity properties on the function it includes a much broader class). I'm speculating here but related ideas in geometry are sometimes called synthetic notions of curvature and that is perhaps some source of confusion (as I understand this term is also used in categorical treatments for something else). | |
| May 12, 2023 at 0:26 | comment | added | Tim Campion | I guess your point is that from a Lorentzian manifold it's easy to construct spaces which still have nice causality relations without actually being smooth? So either the conditions on the poset will have to rule out deformations which look natural order-theoretically, or else they will have to give some "pseudo-smooth structure" to non-manifolds. This seems like an important point. I think I'm tending to think I want notions which will do the latter, but it does look a bit implausible when you put it this way. | |
| May 12, 2023 at 0:25 | comment | added | Tim Campion | I think what I mean by a "sufficiently nice poset" is as described in [3] above -- a bicontinuous poset (see Def 2.10 -- this is actually a notion imported from computer science!) which is globally hyperbolic (i.e. intervals are compact in the interval topology). | |
| May 11, 2023 at 23:00 | history | answered | RBega2 | CC BY-SA 4.0 |