Timeline for Canonical Metric on Grassmann Manifold
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Jan 26, 2016 at 17:47 | comment | added | user21574 | I didn't know the book of Besse, I just checked it. It seems it had long history and I didn't know anything about it. | |
| Jan 26, 2016 at 17:36 | comment | added | Sebastian Goette | @HassanJolany The main point of the paper you mention is the computation of the Tian invariant. Did you find anything on the metric in Besse? To the best of my knowledge, the Kähler-Einstein property for irreducible Hermitian symmetric spaces has been known long before. It follows from curvature computations using the Lie brackets of the Lie groups involved. | |
| Jan 26, 2016 at 16:33 | comment | added | Deane Yang | To repeat what Sebastian already said, the question refers to Grassmannians, which most of us would interpret as real Grassmannians, i.e., $G(n,k) = $ the space of $k$-linear subspaces in an $n$-dimensional real vector space. In general there is no complex structure on such a space, for example if the Grassmannian is odd-dimensional. However, it is a homogeneous space, and the standard meaning of a "canonical metric" on a homogeneous space is one that is invariant under the group action. | |
| Jan 26, 2016 at 16:14 | comment | added | user21574 | Have you seen this paper? jgrivaux.perso.math.cnrs.fr/articles/Tian.pdf In fact we use alpha invariant of Tian to get Kahler Eiinstein metric, when the first Chern class is positive , I am still woundring if you show me it is Kahler Einstein | |
| Jan 26, 2016 at 16:08 | comment | added | Sebastian Goette | @HassanJolany "Canonical" as I understand means "produced by a process involving no additional choices". Since the manifolds here are (probably) real, there is no chance of mistaking "canonical" with something related to a "canonical" bundle or else. But these metrics are symmetric of compact type, so they have many nice properties (Einstein, parallel curvature tensor, hence the sectional curvature has the same structure everywhere, nonnegative curvature operator, so all harmonic differential forms are parallel etc.). What else would you ask for? | |
| Jan 26, 2016 at 15:25 | comment | added | user21574 | what is the your definition about canonical metric,? | |
| Jan 26, 2016 at 12:40 | comment | added | Sebastian Goette | @HassanJolany This metric is actually symmetric and unique up to a constant multiple, as stated above. Any normalisation you wish to choose makes it canonical. | |
| Jan 18, 2016 at 16:26 | comment | added | user21574 | he asking canonical metric, | |
| Aug 15, 2014 at 6:08 | history | edited | Misha Verbitsky | CC BY-SA 3.0 |
added 30 characters in body
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| 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 | history | answered | Misha Verbitsky | CC BY-SA 3.0 |