论文标题
单侧类型D RICCI-FLAT指标
One-sided type-D Ricci-flat metrics
论文作者
论文摘要
我们考虑了四维，riemannian，ricci-flat指标，其中一种或另一种自偶或抗二重性的weyl张量是type-d。这样的指标始终具有Valence-2杀死的旋转器，因此具有遗传学结构和至少一个杀死向量。我们对Przanowski和合作者的结果进行了重新评估，这些指标都可以根据$ su（\ infty）$ -TODA字段方程式来给出，并表明，当有第二个杀戮向量与第一个杀人载体通勤时，可以将病房的方法应用于固定的三分之一的方法。因此，尤其是字段方程线性化。 作为推论，我们表明，相同的技术线性线性线性的三维爱因斯坦度量标准，带有抗二重性Weyl张量和两个通勤对称性。 我们使用非零标量曲率和单侧D型Weyl张量（不包括Kähler-Einstein案例）将爱因斯坦方程式减少到一个修改但不可集成的TODA方程中。 给出了一些构造的例子。
We consider four-dimensional, Riemannian, Ricci-flat metrics for which one or other of the self-dual or anti-self-dual Weyl tensors is type-D. Such metrics always have a valence-2 Killing spinor, and therefore a Hermitian structure and at least one Killing vector. We rederive the results of Przanowski and collaborators, that these metrics can all be given in terms of a solution of the $SU(\infty)$-Toda field equation, and show that, when there is a second Killing vector commuting with the first, the method of Ward can be applied to show that the metrics can also be given in terms of an axisymmetric solution of the flat three-dimensional Laplacian. Thus in particular the field equations linearise. As a corollary, we show that the same technique linearises the field equations for a four-dimensional Einstein metric with anti-self-dual Weyl tensor and two commuting symmetries. We reduce the Einstein equations with non-zero scalar curvature and one-sided type-D Weyl tensor, excluding the Kähler-Einstein case, to a modified but not integrable Toda equation. Some examples of the constructions are given.
