Timeline for How do Lie groups classify geometry?
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4 events
| when toggle format | what | by | license | comment | |
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| Oct 15, 2010 at 19:34 | comment | added | Steve Huntsman | The Berger classification is for irreducible nonsymmetric spaces. Symmetric spaces yield more generic holonomy groups. | |
| Oct 15, 2010 at 15:20 | comment | added | José Figueroa-O'Farrill | Whereas indeed, as Deane points out, this Berger theorem is about riemannian manifolds whose holonomy groups are therefore compact, his results can be extended. First, to the pseudo-riemannian situation: lorentzian holonomy groups (the indecomposable ones necessarily non-compact) have been classified recently and there is a partial classification for index 2. But Berger also worked on the classification of affine torsion-free connections which need not preserve a metric. This classification has also been completed by Merkulov and Schwachhöfer. | |
| Oct 15, 2010 at 13:19 | comment | added | Deane Yang | It seems to me that de Rham and Berger's work is about holonomy groups, which is related to but not the same as the idea of a Lie group defining a geometry (which I interpret as defining the symmetries of a space). In particular, any Lie group is the group of symmetries of a space (namely itself), but Berger's theorem shows that only some Lie groups arise as a holonomy group. Another point is that Berger's theorem applies only to Riemannian manifolds and compact Lie groups, so it misses a lot of important and interesting geometric structures (such as the ones in Thurston's classification). | |
| Oct 15, 2010 at 5:37 | history | answered | Steve Huntsman | CC BY-SA 2.5 |