The most recent version of the theorem, as stated by Nikonorov in 2004(adsbygoogle = window.adsbygoogle || []).push({});

Let G a connected compact semisimple Lie group, which acts almost effectively on a seven-dimensional simply-connected homogeneous space [itex]M^7=G/H[/itex]. If [itex](G/H,\rho)[/itex] 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 [tex]S^7[/tex] being diffeomorphic to [tex]SO(8)/SO(7)[/tex]

s.2 [tex]S^5 \times S^2[/itex] being diff to [tex]{SO(6)\over SO(5)} \times {SO(3)\over SO(2)}[/tex]

s.3 [tex]S^4 \times S^3[/tex] diff to [tex]{SO(5)\over SO(4)} \times {SU(2)\times SU(2) \over SU(2)}[/tex]

s.4 [tex]{SU(3)\over SO(3)} \times S^2[/tex] diff to [tex]{SU(3)\over SO(3)} \times {SO(3)\over SO(2)}[/tex]

s.5 [tex]S^7[/tex] diff to [tex]Spin(7)/G_2[/tex]

(and note also [tex]SU(4)/SU(3)[/tex], which produces again the standard metric on S^7)

and besides we have: -two spaces with a single invariant Einstein metric:

i.1 [tex]Sp(2)/SU(2)[/tex]

i.2 Stiefel [tex]V_{5,3} \equiv SO(5)/SO(3)[/tex]

-two spaces with two metrics available:

i.3 again [tex]S^7[/tex] now as [tex]Sp(2)\over Sp(1)[/tex]

i.5c [tex]T_1S^3 \times S^2[/tex], an special case (1,1,0) of the family 5.

-two families with one metric for each embedding:

i.4 The biparametric (a,b)=[tex]SU(3)\times SU(2) \over SU(2) \times U(1)[/tex]

i.5a the triparametric (a,b,c)=[tex]SU(2) \times SU(2) \times SU(2) \over SO(2) \times SO(2)[/tex]

which has the subfamily of special cases

i.5b (a,b,0) each with a pure factor space [itex]\times S^2[/itex] and then the special case i.5.c above

-and one family with two metrics for each embedding

i.6 the biparametric (a,b)= [tex]SU(3)\over SO(2)[/tex]

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.

Family i.4 is sometime referred as "simply connected Witten spaces". Family i.6 is known to matematicians as Allof-Wallach spaces.

All the three families can be buit as principal [itex]S^1[/itex] fiber bundles:

i.4 over [itex]CP^2 \times S^2[/itex]

i.5 over [itex]S^2 \times S^2 \times S^2[/itex]

i.6 over [itex]SU(3)/T^2[/itex]

All the three families produce groups of isometry greater than g by an U(1) factor.

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# List of Compact 7 dimensional Einstein manifolds

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