Timeline for Geodesics on Homogeneous Spaces of $SU(n)$
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 16, 2017 at 15:47 | vote | accept | Benjamin | ||
| Jan 27, 2017 at 3:07 | comment | added | user21574 | Allen was right, see corollary page 5, cambridge.org/core/services/aop-cambridge-core/content/view/… | |
| Jan 25, 2017 at 5:01 | comment | added | Allen Knutson | A note about @YuryUstinovskiy's comment: the K\"ahler metric on a coadjoint orbit of a compact group is not the metric restricted from $\mathfrak g^*$. Indeed, for $SU(2)$ where the coadjoint orbits are concentric $2$-spheres, the K\"ahler area of an orbit is proportional to its radius, not the square of its radius. | |
| Jan 25, 2017 at 4:28 | comment | added | user21574 | there is a difference between real coadjoint orbit, and complex coadjoint orbit. real coadjoint orbit is affine, but complex coadjoint orbit is projective . see my old question mathoverflow.net/questions/156394/is-g-t-a-projective-variety | |
| Jan 25, 2017 at 4:12 | comment | added | user21574 | Note that, I said complex coadjoint orbit, .ie $G^{\mathbb C}/P$, which is projective variety | |
| Jan 25, 2017 at 0:04 | comment | added | Yury Ustinovskiy | If your $M$ is a coadjoint orbit as in Hassan's comment, then $M$ is a submanifold of $\mathfrak g^*$ and naturally reductive metric is given by restricting the metric $\kappa$ from $\mathfrak g^*$. | |
| Jan 25, 2017 at 0:02 | comment | added | Yury Ustinovskiy | Depends on what you mean by expression. If you have fixed identification $\mathfrak k^\perp\simeq T_pM$, then the naturally reductive metric is given the restriction of the Killing metric, which for SU(n) is just the Frobenius norm. | |
| Jan 24, 2017 at 22:24 | comment | added | Benjamin | Is there an expression for the metric? | |
| Jan 24, 2017 at 21:34 | history | edited | user21574 | CC BY-SA 3.0 |
added 2 characters in body
|
| Jan 24, 2017 at 21:19 | comment | added | YCor | OK I thought you referred to the reductive metric on $SU(n)$. Indeed there's no ambiguity on $SU(n)/K$ (if one still refers to the reductive metric). | |
| Jan 24, 2017 at 21:05 | comment | added | Yury Ustinovskiy | Sorry, I am not sure I got your point. I am talking about invariant metrics on homogeneous spaces, and they admit only left group action, unless it is some very special case, like $M=SU(n)$ | |
| Jan 24, 2017 at 20:50 | comment | added | YCor | in your last sentence, you probably mean left-invariant? the bi-invariant one is unique up to scalar multiplication. "invariant" means both left or bi-invariant according to authors. | |
| Jan 24, 2017 at 20:23 | history | answered | Yury Ustinovskiy | CC BY-SA 3.0 |