Timeline for Visualizing Bianchi type/homogenous spaces
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Mar 17, 2013 at 17:20 | vote | accept | duetosymmetry | ||
| Mar 17, 2013 at 5:00 | answer | added | Robert Bryant | timeline score: 8 | |
| Mar 16, 2013 at 1:17 | comment | added | Ryan Budney | @duetosymmetry: the Thurston notes very much goes into the geometry of these spaces. Or perhaps you should specify what you mean by "geometry". If "geometry" means "spectrum of the laplacian" or something like that, then sure, the Thurston notes don't talk about that kind of thing. | |
| Mar 16, 2013 at 1:15 | comment | added | duetosymmetry | For two dimensional homogenous spaces, I can visualize the embedding in $E^3$ as the standard unit sphere or the hyperbolic plane. I imagine that in some 3-dimensional homogenous spaces, there may be a natural foliation by 2-manifolds, and I could consider the induced geometry on a sequence of 2-manifolds and look at a sequence of embeddings. How do I know what I'd see? | |
| Mar 16, 2013 at 1:07 | history | edited | duetosymmetry | CC BY-SA 3.0 |
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| Mar 16, 2013 at 0:42 | comment | added | Robert Bryant | It's not clear what you want to 'see' in your visualization. For example, there is a 3-parameter family of mutually inequivalent left-invariant metrics on $S^3$ (regarded as a Lie group), and these can be viewed as perturbations of the constant positive curvature metric. At least for small perturbations, they will 'look' very much like the constant curvature metric, though the geodesics leaving a point won't all focus on a single point when they reach the 'other side', they'll just get close together before they spread out again. Do you find that helpful for visualization? More like this? | |
| Mar 15, 2013 at 21:36 | comment | added | Ryan Budney | There's Scott's paper "The geometries of 3-manifolds" math.lsa.umich.edu/~pscott as well as Thurston's book "Three-dimensional geometry and topology" that are both quite good primers on this topic. I've never read a Bianchi paper so I don't know how his "types" correspond to the geometries I'm familiar with. | |
| Mar 15, 2013 at 21:24 | history | asked | duetosymmetry | CC BY-SA 3.0 |