Anti-Self-Dual Metrics and Kähler Geometry
Anti-Self-Dual Metrics and Kähler Geometry
复制标题
反自对偶度量和凯勒几何
DOI：
--
复制
发表时间：
1995
期刊：
影响因子：
--
通讯作者：
C. LeBrun
中科院分区：
文献类型：
--
作者：
C. LeBrun
李群SO（4）是非单的这一事实赋予了四维几何一种极其独特的味道。实际上，在定向4-流形M上选择黎曼度量g分裂了2-形式丛
$${Lambda ^2} = {Lambda ^ + } o加上{Lambda ^ - }$$
（一）
分别定义为Hodge星星算子的±1-特征空间：Hodge 2 → Hodge 2;这正好反映了SO（4）在斜（4 × 4）-矩阵上的伴随表示是两个三维表示之和的事实，如李代数同构so（4）Hodge so（3）Hodge so（3）Hodge so（4）.分解（1）是共形不变的，即对于任何正函数u，如果g被ug代替，它是不变的;但是反转M的方向，则交换了丛<$±。
The fact that the Lie group SO(4) is nonsimple gives 4-dimensional geometry an extremely distinctive flavor. Indeed, the choice of a Riemannian metric g on an oriented 4-manifold M splits the bundle of 2-forms
$${Lambda ^2} = {Lambda ^ + } oplus {Lambda ^ - }$$
(1)
into the rank-3 bundles of self-dual and anti-self dual 2-forms, respectively defined as the ±1-eigenspaces of the Hodge star operator ⋆ : ⋀2 → ⋀2; this just reflects the fact that the adjoint representation of SO(4) on the skew (4 x 4)-matrices is the sum of two 3-dimensional representations, as indicated by the Lie algebra isomorphism so(4) ≅ so(3)⊕so(3). The decomposition (1) is conformally invariant, in the sense that it is unchanged if g is replaced by ug for any positive function u; but reversing the orientation of M interchanges the bundles ⋀±.