标题： 旗流形上齐次爱因斯坦度量的分类与$b_2(M)=1$ 显示英文标题

标题： The classification of homogeneous Einstein metrics on flag manifolds with $b_2(M)=1$

摘要： 设$G$为一个单紧致连通李群。 我们研究一类紧致齐次空间的齐次爱因斯坦度量，即第二贝蒂数为$b_{2}(G/H)=1$的广义旗流形$G/H$。 有8个无限族$G/H$对应于一个经典单李群$G$，以及25个例外旗流形，它们都有一些共同的几何特征；例如它们都具有唯一的不变复结构，该结构产生唯一的不变凯勒-爱因斯坦度量。 最典型的例子是紧致不可约的全纯对称空间，其卡林根形式是唯一的齐次爱因斯坦度量（该度量为凯勒）。 对于不可约不可约空间，齐次爱因斯坦度量的分类已完成了26种情况中的24种。 在本文中，我们为另外两种未检查的情况构建爱因斯坦方程，即旗流形$\E_8/\U(1)\times \SU(4)\times \SU(5)$和$\E_8/\U(1)\times \SU(2)\times \SU(3)\times \SU(5)$。 为了明确确定$\E_8$-不变度量的里奇张量，我们使用基于黎曼子流形的方法。 对于这两个空间，我们分类了所有齐次爱因斯坦度量，因此得出结论：任何旗流形$G/H$与$b_{2}(M)=1$都具有有限数量的非同构的非凯勒不变爱因斯坦度量。 这些度量的确切数量在表1中给出。 显示英文摘要

摘要： Let $G$ be a simple compact connected Lie group. We study homogeneous Einstein metrics for a class of compact homogeneous spaces, namely generalized flag manifolds $G/H$ with second Betti number $b_{2}(G/H)=1$. There are 8 infinite families $G/H$ corresponding to a classical simple Lie group $G$ and 25 exceptional flag manifolds, which all have some common geometric features; for example they admit a unique invariant complex structure which gives rise to unique invariant K\"ahler--Einstein metric. The most typical examples are the compact isotropy irreducible Hermitian symmetric spaces for which the Killing form is the unique homogeneous Einstein metric (which is K\"ahler). For non-isotropy irreducible spaces the classification of homogeneous Einstein metrics has been completed for 24 of the 26 cases. In this paper we construct the Einstein equation for the two unexamined cases, namely the flag manifolds $\E_8/\U(1)\times \SU(4)\times \SU(5)$ and $\E_8/\U(1)\times \SU(2)\times \SU(3)\times \SU(5)$. In order to determine explicitly the Ricci tensors of an $\E_8$-invariant metric we use a method based on the Riemannian submersions. For both spaces we classify all homogeneous Einstein metrics and thus we conclude that any flag manifold $G/H$ with $b_{2}(M)=1$ admits a finite number of non-isometric non-K\"ahler invariant Einstein metrics. The precise number of these metrics is given in Table 1.