Timeline for Fréchet manifolds vs ILH manifolds
Current License: CC BY-SA 3.0
7 events
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| Aug 31, 2011 at 9:20 | comment | added | Andrew Stacey | From what you say, I would suspect that if an ILH-group is a manifold then it is a Frechet manifold, but it might not be a manifold. I would need to look much more carefully at the definitions to be sure. | |
| Aug 30, 2011 at 21:29 | comment | added | Igor Belegradek | cont... Later on the same page Omori says: "as far as concerning about ILH Lie groups the inverse limit $G=\lim G^s$ might not be a Frechet manifold (and probably there is a counterexample). On the other hand, the diffeomorphism group of a closed manifold is not only an ILH Lie group and a Frechet Lie group but also have some nice properties mentioned above." My take is that the strong ILH Lie group is a Frechet manifold, but this is not true if strong is dropped. | |
| Aug 30, 2011 at 21:28 | comment | added | Igor Belegradek | I know two Omori's books: "Infinite dimensional Lie transformation groups" (LNM 427) and "Infinite dimensional Lie groups" (translations of Mathematical monographs, 158. In the former one on page 3 he defines an ILH group $G$ as the inverse limit of topological groups $G^s$ that satisfy seven technical conditions. | |
| Aug 30, 2011 at 20:55 | comment | added | Andrew Stacey | Which book of Omori's are you looking at? I thought that there was a nice exposition in one of them, but I may be misremembering. That last sentence that you say sounds familiar, certainly. | |
| Aug 30, 2011 at 18:54 | comment | added | Igor Belegradek | Kriegl-Michor's book doesn't discuss ILH structures; instead they refer to Omori's book, which I think doesn't deal with the trival matters I am concerned with. I like the presentation in Omori's paper "On the group of diffeomorphisms of a compact manifold" in Global Analysis 1970, where he mentions in passing that a strong ILH manifold is a Frechet manifold whose transition functions are linear isomorphisms of the model space that extend to linear isomorphisms of the Hilbert spaces "converging" to the model space, and moreover this structure is integrable. | |
| Aug 30, 2011 at 18:02 | comment | added | Andrew Stacey | @Igor: No problem! I'm sorry that it's so vague on the details, but hopefully it's enough for you to find the right details in Omori's books (or in other work, such as Kriegl and Michor's book). | |
| Aug 30, 2011 at 15:18 | history | answered | Andrew Stacey | CC BY-SA 3.0 |