Timeline for Canonical Metric on Grassmann Manifold
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Jan 26, 2016 at 21:14 | comment | added | Robert Bryant | @SebastianGoette: It's a question of terminology, I suppose, and I am not sure what office is qualified to rule on the 'official' notions. `Canonical' in mathematical English has several different meanings; I usually use 'canonical' to mean 'invariantly defined', and when there is more than one 'natural' candidate, I prefer not to call any of them canonical. (By the way, I learned to call it 'Jordan normal form', not 'Jordan canonical form', which, although I see it in various places, always looks a little odd to me, since it's not clear to me just what is canonical about Jordan normal form.) | |
| Jan 26, 2016 at 19:36 | comment | added | Sebastian Goette | @RobertBryant Is there an official notion of a "canonical metric"? Though no metric is invariant under $\mathrm{Aut}(V)$, all metrics produced by some of the extra choices you name give isometric Riemannian manifolds in the end. In my understanding, this is as canonical as one can ever get (compare "Jordan canonical form" etc). | |
| Jan 18, 2016 at 16:16 | comment | added | user21574 | @PaulSiegel , Aaron asked canonical metric not natural metric, ;) . See my answer | |
| Aug 30, 2014 at 7:57 | answer | added | Peter Michor | timeline score: 7 | |
| Aug 30, 2014 at 3:50 | answer | added | Renato G. Bettiol | timeline score: 5 | |
| Aug 15, 2014 at 11:18 | comment | added | Robert Bryant | The space $\mathrm{Gr}(p,V)$ of $p$-dimensional subspaces in a vector space $V$ does not have a nontrivial canonical metric. There is no Riemannian metric on this space that is invariant under the natural action of $\mathrm{Aut}(V)=\mathrm{GL}(V)$. Upon fixing an additional structure on $V$, namely, a positive definite inner product $q$, there is a Riemannian metric on this space that is invariant under the natural action of $\mathrm{Aut}(V,q)=\mathrm{O}(q)$, unique up to a constant multiple. This multiple can be uniquely determined by specifying the total volume or diameter, for example. | |
| Aug 15, 2014 at 3:04 | vote | accept | Aaron Oberländer | ||
| Aug 15, 2014 at 3:03 | vote | accept | Aaron Oberländer | ||
| Aug 15, 2014 at 3:04 | |||||
| Aug 15, 2014 at 2:54 | answer | added | Misha Verbitsky | timeline score: 14 | |
| Aug 15, 2014 at 2:33 | comment | added | user62675 | See montefiore.ulg.ac.be/systems/Publi/Grass_geom.pdf, for the Grassmann manifold of $p$-planes in $\mathbf{R}^n$. | |
| Aug 15, 2014 at 1:46 | comment | added | David E Speyer | If you want an explicit formula, see mathoverflow.net/questions/141483/… | |
| Aug 14, 2014 at 23:28 | comment | added | Paul Siegel | The Grassmanian is a homogeneous space for the orthogonal group (unitary group in the complex case) and hence inherits a natural metric. | |
| Aug 14, 2014 at 23:13 | review | First posts | |||
| Aug 14, 2014 at 23:32 | |||||
| Aug 14, 2014 at 23:05 | history | asked | Aaron Oberländer | CC BY-SA 3.0 |