Distinguished dimensions for special Riemannian geometries
Distinguished dimensions for special Riemannian geometries
复制标题
DOI：
10.1016/j.geomphys.2008.03.012
复制
发表时间：
2006-01
影响因子：
1.5
通讯作者：
P. Nurowski
中科院分区：
文献类型：
--
作者：
P. Nurowski
本文基于定义五维SO(3)几何的三元对称形式与Cartan关于球面等参超曲面的工作之间的关系。正如布莱恩特所观察到的，这种三元形式只存在于维度nk=3k+2中，其中k=1，2，4，8。在这些维度中，它将正交群缩减为子群HK⊂So(Nk)，其中H1=S0(3)，H2=SU(3)，H4=Sp(3)和H8=F4。这使得研究具有结构群Hkin维度nk的特殊黎曼几何成为可能。给出了Hk几何允许特征联络的充要条件。作为说明，给出了8维SU(3)几何允许特征联系的非平凡例子。其中有关于李维-西维塔或特征联系具有非零挠率且满足爱因斯坦方程的例子。Hk几何的无扭转模型有各自的对称群G1=SU(3)，G2=SU(3)×SU(3)，G3=SU(6)和G4=E6。群Hk和Gk构成了李群的‘幻方’的一部分。‘幻方’李群建议研究其他十类特殊的黎曼几何。除了两种例外情况外，它们还有结构群U(3)、S(U(3)×U(3))、U(6)、E6×SO(2)、Sp(3)×SU(2)、SU(6)×SU(2)、SO(12)×SU(2)和E7×SU(2)，应分别在12、18、30、54、28、40、和112维中考虑。两种“例外”情况是：8维的SU(2)×SU(2)几何和32维的SO(10)×SO(2)几何。进一步研究了8维空间中SU(2)×SU(2)几何的情形。我们确定了将SO(8)简化为SU(2)×SU(2)的张量，将基于幻方思想的几何的更详细的研究留到即将发表的论文中。
The paper is based on relations between a ternary symmetric form defining the SO(3) geometry in dimension five and Cartan’s works on isoparametric hypersurfaces in spheres. As observed by Bryant such a ternary form exists only in dimensions nk=3k+2, where k=1,2,4,8. In these dimensions it reduces the orthogonal group to the subgroups Hk⊂SO(nk), with H1=SO(3), H2=SU(3), H4=Sp(3) and H8=F4. This enables studies of special Riemannian geometries with structure groups Hkin dimensions nk. The necessary and sufficient conditions for the Hkgeometries to admit the characteristic connection are given. As an illustration nontrivial examples of SU(3) geometries in dimension 8 admitting characteristic connection are provided. Among them are the examples having nonvanishing torsion and satisfying Einstein equations with respect to either the Levi-Civita or the characteristic connections. The torsionless models for the Hkgeometries have the respective symmetry groups G1=SU(3), G2=SU(3)×SU(3), G3=SU(6) and G4=E6. The groups Hkand Gkconstitute a part of the ‘magic square’ for Lie groups. The ‘magic square’ Lie groups suggest studies of ten other classes of special Riemannian geometries. Apart from the two exceptional cases, they have the structure groups U(3), S(U(3)×U(3)), U(6), E6×SO(2), Sp(3)×SU(2), SU(6)×SU(2), SO(12)×SU(2) and E7×SU(2) and should be considered in respective dimensions 12, 18, 30, 54, 28, 40, 64 and 112. The two ‘exceptional’ cases are: SU(2)×SU(2) geometries in dimension 8 and SO(10)×SO(2) geometries in dimension 32. The case of SU(2)×SU(2) geometry in dimension 8 is examined closer. We determine the tensor that reduces SO(8) to SU(2)×SU(2) leaving the more detailed studies of the geometries based on the magic square ideas to the forthcoming paper.