Timeline for How do Lie groups classify geometry?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 16, 2010 at 14:20 | vote | accept | CommunityBot | ||
| Oct 15, 2010 at 14:25 | answer | added | Peter Arndt | timeline score: 5 | |
| Oct 15, 2010 at 14:18 | answer | added | Qfwfq | timeline score: 3 | |
| Oct 15, 2010 at 13:42 | comment | added | Tim Perutz | Taking up Andy's point, I think your question mixes three distinct ideas. (1) Pseudogroups. (2) Homogeneous spaces and their symmetry groups, Erlangen program, geometric structures on manifolds in Thurston's sense. (3) The structure group of the tangent bundle. The bundle of tangent frames on a smooth $n$-manifold is a principal $GL(n,\mathbb{R})$-bundle, but e.g. a choice of orientation and Riemannian metric gives a reduction of the structure group to $SO(n)$ (namely, the bundle of orthonormal oriented frames). | |
| Oct 15, 2010 at 5:37 | answer | added | Steve Huntsman | timeline score: 4 | |
| Oct 15, 2010 at 5:14 | comment | added | Andy Putman | Defining structures on mnflds using pseudogroups of morphisms is a very general technique, and the pseudogroups of morphisms don't all need to come from Lie groups (eg a smooth manifold uses the pseudogroup of smooth diffeomorphisms between open subsets of Euclidean spaces, a foliated manifold uses the pseudogroup of diffeomorphisms between open subsets of Euclidean spaces that can be expressed in the form $\phi(x,y)=(\phi_1(x,y),\phi_2(y))$, etc.). Probably the best way to get introduced to this point of view is to read Chapter 3 of Thurston's book "Three-Dimensional Geometry and Topology". | |
| Oct 15, 2010 at 4:56 | history | asked | user2529 | CC BY-SA 2.5 |