论文标题
$ \ mathbb {m}^{1,3} $中的偏斜成符号内态性：一种统一的规范形式
Skew-symmetric endomorphisms in $\mathbb{M}^{1,3}$: A unified canonical form with applications to conformal geometry
论文作者
论文摘要
我们在lorentzian vector vector Spaces的第三和四个偏斜的内态性内态偏斜的内态$ f $ $ f $中得出了一种规范形式，该形式一次涵盖了所有非平凡的情况。我们分析了其不变性组，以及这种规范形式与两种形式的双重旋转的连接。在回顾了这些内态性与$ \ mathbb {s}^2 $，$ \ mathrm {ckill}（\ Mathbb {s}^2）$的保形杀伤量的代数之间的关系之后\ mathrm {ckill}（\ Mathbb {s}^2）$及其不变性组。该构建使我们能够明确获得将任何给定$ f $转换为规范形式的基础的变化。对于任何非平凡的$ξ$，我们通过其规范形式构建的适应坐标，使我们能够深入研究其属性。解决了两个应用程序：我们明确确定哪些指标，在恒定弯曲的天然空间中，给定的$ξ$是杀伤向量，并解决了满足杀死初始数据方程的所有局部TT（无可怜和横向）张量。除了自己的利益外，目前的结果将是随后对任意维度的概括的基本要素。
We derive a canonical form for skew-symmetric endomorphisms $F$ in Lorentzian vector spaces of dimension three and four which covers all non-trivial cases at once. We analyze its invariance group, as well as the connection of this canonical form with duality rotations of two-forms. After reviewing the relation between these endomorphisms and the algebra of conformal Killing vectors of $\mathbb{S}^2$, $\mathrm{CKill}(\mathbb{S}^2)$, we are able to also give a canonical form for an arbitrary element $ ξ\in \mathrm{CKill}(\mathbb{S}^2)$ along with its invariance group. The construction allows us to obtain explicitly the change of basis that transforms any given $F$ into its canonical form. For any non-trivial $ ξ$ we construct, via its canonical form, adapted coordinates that allow us to study its properties in depth. Two applications are worked out: we determine explicitly for which metrics, among a natural class of spaces of constant curvature, a given $ξ$ is a Killing vector and solve all local TT (traceless and transverse) tensors that satisfy the Killing Initial Data equation for $ξ$. In addition to their own interest, the present results will be a basic ingredient for a subsequent generalization to arbitrary dimensions.
