Einstein metrics on group manifolds and cosets
Einstein metrics on group manifolds and cosets
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DOI：
10.1016/j.geomphys.2011.01.004
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发表时间：
2009-03
期刊：
影响因子：
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通讯作者：
G. Gibbons;Hong Lu;Hong Lu;C. Pope;C. Pope
中科院分区：
文献类型：
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作者：
G. Gibbons;Hong Lu;Hong Lu;C. Pope;C. Pope
众所周知，每个紧单群流形 G 都承认一个双不变爱因斯坦度量，在 GL×GR 下不变。不太为人所知的是，除了 SO(3) 和 SU(2) 之外，每个紧单群流形都至少承认一个更齐次的爱因斯坦度量，在 GL 下仍然不变，但部分或全部右作用对称性被破坏。 （SO(3) 和 SU(2) 的例外之处在于只接受一个双不变的爱因斯坦度量。）在本文中，我们在三个相对低维的例子上寻找爱因斯坦度量，即 G=SU(3)、SO(5) 和 G2。对于 G=SU(3)，我们只找到两个已知的不等价爱因斯坦度量。对于 G=SO(5)，我们发现了四个不等价的爱因斯坦度量，从而扩展了之前仅已知两个的结果。对于 G=G2，我们找到了六个不等价的爱因斯坦度量，这将列表扩展到了先前已知的两个示例之外。我们还研究了上述群 G 的一些陪集 G/H。特别是，对于 SO(5)/U(1)，我们发现，根据 U(1) 的嵌入，通常有两个，特别是一到三个爱因斯坦度量。我们还发现了 SU(3) 上签名 (2,6) 的伪黎曼爱因斯坦度量、G2/SU(2)diag 上签名 (5,6) 的爱因斯坦度量以及 G2/U(2) 上签名 (4,6) 的爱因斯坦度量。有趣的是，我们的例子中没有洛伦兹爱因斯坦度量。
It is well known that every compact simple group manifold G admits a bi-invariant Einstein metric, invariant under GL×GR. Less well known is that every compact simple group manifold except SO(3) and SU(2) admits at least one more homogeneous Einstein metric, invariant still under GLbut with some, or all, of the right-acting symmetry broken. (SO(3) and SU(2) are exceptional in admitting only the one, bi-invariant, Einstein metric.) In this paper, we look for Einstein metrics on three relatively low-dimensional examples, namely G=SU(3), SO(5) and G2. For G=SU(3), we find just the two already known inequivalent Einstein metrics. For G=SO(5), we find four inequivalent Einstein metrics, thus extending previous results where only two were known. For G=G2we find six inequivalent Einstein metrics, which extends the list beyond the previously-known two examples. We also study some cosets G/H for the above groups G. In particular, for SO(5)/U(1) we find, depending on the embedding of the U(1), generically two, with exceptionally one or three, Einstein metrics. We also find a pseudo-Riemannian Einstein metric of signature (2,6) on SU(3), an Einstein metric of signature (5,6) on G2/SU(2)diag, and an Einstein metric of signature (4,6) on G2/U(2). Interestingly, there are no Lorentzian Einstein metrics among our examples.