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arivero
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The most recent version of the theorem, as stated by Nikonorov in 2004
Let G a connected compact semisimple Lie group, which acts almost effectively on a seven-dimensional simply-connected homogeneous space M^7=G/H. If (G/H,\rho) is a homogeneous Einstein manifold, then it is either a symmetric space or isometric to one of the manifolds in i.1 to i.6
These are the spaces having each exactly one Einstein metric, the symmetric one:
s.1 S^7 being diffeomorphic to SO(8)/SO(7)
s.2 S^5 \times S^2[/itex] being diff to {SO(6)\over SO(5)} \times {SO(3)\over SO(2)}<br /> <br /> s.3 S^4 \times S^3 diff to {SO(5)\over SO(4)} \times {SU(2)\times SU(2) \over SU(2)}<br /> <br /> s.4 {SU(3)\over SO(3)} \times S^2 diff to {SU(3)\over SO(3)} \times {SO(3)\over SO(2)}<br /> <br /> s.5 S^7 diff to Spin(7)/G_2<br /> <br /> (and note also SU(4)/SU(3), which produces again the standard metric on S^7)<br /> <br /> and besides we have: -two spaces with a single invariant Einstein metric:<br /> <br /> i.1 Sp(2)/SU(2)<br /> <br /> i.2 Stiefel V_{5,3} \equiv SO(5)/SO(3)<br /> <br /> -two spaces with two metrics available:<br /> <br /> i.3 again S^7 now as Sp(2)\over Sp(1)<br /> <br /> i.5c T_1S^3 \times S^2, an special case (1,1,0) of the family 5.<br /> <br /> -two families with one metric for each embedding:<br /> <br /> i.4 The biparametric (a,b)=SU(3)\times SU(2) \over SU(2) \times U(1)<br /> <br /> i.5a the triparametric (a,b,c)=SU(2) \times SU(2) \times SU(2) \over SO(2) \times SO(2)<br /> which has the subfamily of special cases<br /> i.5b (a,b,0) each with a pure factor space \times S^2 and then the special case i.5.c above<br /> <br /> -and one family with two metrics for each embedding<br /> <br /> i.6 the biparametric (a,b)= SU(3)\over SO(2)<br /> <br /> Note I am abusing the notation of the parameters because the table in the theorem does not follow previous notations. This is a tradition in the field, it seems.<br /> <br /> Family i.4 is sometime referred as "simply connected Witten spaces". Family i.6 is known to matematicians as Allof-Wallach spaces.<br /> <br /> All the three families can be buit as principal S^1 fiber bundles:<br /> i.4 over CP^2 \times S^2<br /> i.5 over S^2 \times S^2 \times S^2<br /> i.6 over SU(3)/T^2<br /> <br /> All the three families produce groups of isometry greater than g by an U(1) factor.
Let G a connected compact semisimple Lie group, which acts almost effectively on a seven-dimensional simply-connected homogeneous space M^7=G/H. If (G/H,\rho) is a homogeneous Einstein manifold, then it is either a symmetric space or isometric to one of the manifolds in i.1 to i.6
These are the spaces having each exactly one Einstein metric, the symmetric one:
s.1 S^7 being diffeomorphic to SO(8)/SO(7)
s.2 S^5 \times S^2[/itex] being diff to {SO(6)\over SO(5)} \times {SO(3)\over SO(2)}<br /> <br /> s.3 S^4 \times S^3 diff to {SO(5)\over SO(4)} \times {SU(2)\times SU(2) \over SU(2)}<br /> <br /> s.4 {SU(3)\over SO(3)} \times S^2 diff to {SU(3)\over SO(3)} \times {SO(3)\over SO(2)}<br /> <br /> s.5 S^7 diff to Spin(7)/G_2<br /> <br /> (and note also SU(4)/SU(3), which produces again the standard metric on S^7)<br /> <br /> and besides we have: -two spaces with a single invariant Einstein metric:<br /> <br /> i.1 Sp(2)/SU(2)<br /> <br /> i.2 Stiefel V_{5,3} \equiv SO(5)/SO(3)<br /> <br /> -two spaces with two metrics available:<br /> <br /> i.3 again S^7 now as Sp(2)\over Sp(1)<br /> <br /> i.5c T_1S^3 \times S^2, an special case (1,1,0) of the family 5.<br /> <br /> -two families with one metric for each embedding:<br /> <br /> i.4 The biparametric (a,b)=SU(3)\times SU(2) \over SU(2) \times U(1)<br /> <br /> i.5a the triparametric (a,b,c)=SU(2) \times SU(2) \times SU(2) \over SO(2) \times SO(2)<br /> which has the subfamily of special cases<br /> i.5b (a,b,0) each with a pure factor space \times S^2 and then the special case i.5.c above<br /> <br /> -and one family with two metrics for each embedding<br /> <br /> i.6 the biparametric (a,b)= SU(3)\over SO(2)<br /> <br /> Note I am abusing the notation of the parameters because the table in the theorem does not follow previous notations. This is a tradition in the field, it seems.<br /> <br /> Family i.4 is sometime referred as "simply connected Witten spaces". Family i.6 is known to matematicians as Allof-Wallach spaces.<br /> <br /> All the three families can be buit as principal S^1 fiber bundles:<br /> i.4 over CP^2 \times S^2<br /> i.5 over S^2 \times S^2 \times S^2<br /> i.6 over SU(3)/T^2<br /> <br /> All the three families produce groups of isometry greater than g by an U(1) factor.
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