论文标题
非kählerc空间上的同质爱因斯坦指标
Homogeneous Einstein metrics on non-Kähler C-spaces
论文作者
论文摘要
我们研究了同质的爱因斯坦指标，这些指标在不可塑性的非kählerianc空间上，即均值维的圆环捆绑包$ m = g/h $，带有$ \ mathsf {rank} g> \ mathsf {ranksf {rank rank} h $ avel flag compact $ f = g = g/k compact compact compact lie lie lie clipe $ g $。基于涂漆的Dynkin图理论，我们介绍了此类空间的分类。接下来，我们将重点放在家庭上\ [m _ {\ ell，m，n}：= \ Mathsf {su}（\ ell+m+n）/\ Mathsf {susf {su}（\ ell）\ times \ times \ times \ times \ mathsf {su}（m） n \ in \ mathbb {z} _ {+} \]并检查其几何属性。我们表明，$ m _ {\ ell，m，n} $上的不变指标不是对角线的，除了某些例外，它们的参数化取决于六个实际参数。通过使用这种不变的riemannian度量，我们计算了ricci张量的对角线和非对角线部分，并明确地呈现同质爱因斯坦方程的代数系统。对于一般的积极整数$ \ ell，m，n $，通过应用映射学位理论，我们提供了至少一个$ \ mathsf {su}（\ ell+m+m+n）$ - 不变的爱因斯坦公制，$ m _ {\ ell，m，n} $。 For $\ell=m$ we show the existence of two $\mathsf{SU}(2m+n)$ invariant Einstein metrics on $M_{m, m, n}$, and for $\ell=m=n$ we obtain four $\mathsf{SU}(3n)$-invariant Einstein metrics on $M_{n, n, n}$.我们还检查了这些指标的等轴测问题，而对于固定的$ \ ell，m，n $引起的大量病例，我们提供了所有非等法不变的爱因斯坦指标的数值形式。
We study homogeneous Einstein metrics on indecomposable non-Kählerian C-spaces, i.e. even-dimensional torus bundles $M=G/H$ with $\mathsf{rank} G>\mathsf{rank} H$ over flag manifolds $F=G/K$ of a compact simple Lie group $G$. Based on the theory of painted Dynkin diagrams we present the classification of such spaces. Next we focus on the family \[ M_{\ell, m, n}:=\mathsf{SU}(\ell+m+n)/\mathsf{SU}(\ell)\times\mathsf{SU}(m)\times\mathsf{SU}(n)\,,\quad \ell, m, n\in\mathbb{Z}_{+} \] and examine several of its geometric properties. We show that invariant metrics on $M_{\ell, m, n}$ are not diagonal and beyond certain exceptions their parametrization depends on six real parameters. By using such an invariant Riemannian metric, we compute the diagonal and the non-diagonal part of the Ricci tensor and present explicitly the algebraic system of the homogeneous Einstein equation. For general positive integers $\ell, m, n$, by applying mapping degree theory we provide the existence of at least one $\mathsf{SU}(\ell+m+n)$-invariant Einstein metric on $M_{\ell, m, n}$. For $\ell=m$ we show the existence of two $\mathsf{SU}(2m+n)$ invariant Einstein metrics on $M_{m, m, n}$, and for $\ell=m=n$ we obtain four $\mathsf{SU}(3n)$-invariant Einstein metrics on $M_{n, n, n}$. We also examine the isometry problem for these metrics, while for a plethora of cases induced by fixed $\ell, m, n$, we provide the numerical form of all non-isometric invariant Einstein metrics.
