Timeline for Notion of smoothness for set-valued functions
Current License: CC BY-SA 2.5
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 9, 2010 at 13:09 | vote | accept | Willie Wong | ||
| Sep 9, 2010 at 0:54 | comment | added | Deane Yang | Another way to think about this is that the cone $K$ can be specified by $K = \{ (x,\xi), h(x,\xi) \ge 0\}$, where $h: T^*M \rightarrow \R$ is a function where $h(x, t\xi) = th(x,\xi)$ for each $(x,\xi) \in T^*M$ and $t > 0$. Then you would call the cone smooth if $h$ can be chosen to be a smooth function and the fiber gradient of $h$ is nonzero when $h = 0$. This is, I believe, the same definition as Sergei's and Tom's. | |
| Sep 8, 2010 at 19:09 | answer | added | Peter Arndt | timeline score: 4 | |
| Sep 8, 2010 at 18:19 | comment | added | Willie Wong | @Sergei: hum, that's a thought. Instead of $TM$ I can probably take the bundle of directions. The projection of the cones to that have smooth boundaries. Let me think about that for a bit. | |
| Sep 8, 2010 at 18:17 | comment | added | Willie Wong | @Sergei: a cone by definition has a vertex at the origin, so the boundaries cannot be smooth (except when the cone is a half space). But if you are willing to overlook that one point, the boundary is smooth everywhere else. | |
| Sep 8, 2010 at 18:15 | comment | added | Willie Wong | @Deane: unfortunately, what I can use it for depends a lot on what reasonable definitions can I have. =) Originally I was hoping to be able to have a linear connection. But Michael's comment made me realise that goal is rather unfeasible in the general case. But a spray structure may still be possible (I hope). So one possible interpretation of the question would be: is there a suitable definition of "regularity" for the collection of tangent cones such that it can guarantee a geodesic spray compatible with the cones. | |
| Sep 8, 2010 at 18:10 | comment | added | Sergei Ivanov | In your application, are your cones' boundaries are smooth? If so, you can say that the map is smooth if the union of these boundaries is a smooth submanifold of $TM$. This is consistent with smoothness in Finsler geometry (where one can say that Finsler metric = family of norms = family of convex bodies in the fibers of the tangent bundle). | |
| Sep 8, 2010 at 17:42 | answer | added | Kevin H. Lin | timeline score: 1 | |
| Sep 8, 2010 at 17:08 | comment | added | Todd Trimble | If you are willing to consider highly abstract approaches, you might consider certain toposes of smooth spaces, as developed in say the book by Moerdijk and Reyes. In such toposes one can interpret "the smooth space of smooth subspaces of $Y$", call it $P(Y)$, and then consider smooth maps $f: X \to P(Y)$. It seems possible that the specific types of $f$ you are after would be definable as smooth maps in a great many such toposes. (Unfortunately, I am not a specialist in synthetic differential geometry.) | |
| Sep 8, 2010 at 17:03 | comment | added | Deane Yang | Willie, it seems to me that the "right" definition depends a lot on what you need this for. | |
| Sep 8, 2010 at 16:32 | answer | added | Tom LaGatta | timeline score: 4 | |
| Sep 8, 2010 at 15:26 | comment | added | Willie Wong | But I thought local triviality requires, at the very least, some notion of isomorphism between the fibres $\pi^{-1}(p)$. While it is possible to forget about the linear structure and just use diffeomorphism of manifolds with boundaries as the isomorphism upstairs, I hesitate to do so because it'd be a bit annoying if, after trivializing and picking a coordinate, I cannot "add" two sections in a reasonable way. (Whereas one can do so "upstaris") | |
| Sep 8, 2010 at 15:12 | comment | added | Michael Bächtold | but you can forget about the linear structure and just treat the tangent bundle as a fiber bundle. | |
| Sep 8, 2010 at 15:09 | history | edited | Willie Wong | CC BY-SA 2.5 |
Clarified what a convex cone is.
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| Sep 8, 2010 at 14:55 | comment | added | Willie Wong | I thought about that, but ran into the following problem: for a smooth vector subbundle $D\subset F$, local triviality can be implemented by $D$ being pointwise a vector subspace, which is compatible with the linear structure on $F$. In my case, ideally if I use a condition like that it should be compatible with the linear structure in some way, but one cannot always map an open convex cone to another using a linear map (think one of them being the open half space). Which is why I am trying to get at something using less structure. | |
| Sep 8, 2010 at 14:33 | answer | added | rpotrie | timeline score: 2 | |
| Sep 8, 2010 at 14:29 | comment | added | Michael Bächtold | In your particular example it seems that the usual condition of local triviality (like in smooth sub-bundles) might be an option. | |
| Sep 8, 2010 at 14:14 | history | asked | Willie Wong | CC BY-SA 2.5 |