We show the existence of families of orthonormal, future directed bases which allow to cast every skew-symmetric endomorphism of () in a single canonical form depending on a minimal number of parameters. This canonical form is shared by every pair of elements in differing by an orthochronous Lorentz transformation, i.e. it defines the orbits of the orthochronous Lorentz group O ⁺(1, n) under the adjoint action on its algebra. Using this form, we obtain the quotient topology of . From known relations between and the conformal Killing vector fields (CKVFs) of the sphere , a canonical form for CKVFs follows immediately. This form is used to find adapted coordinates to an arbitrary CKVF that covers all cases at the same time. We do the calculation for even n and obtain the case of odd n as a consequence. Finally, we employ the adapted coordinates to obtain a wide class of TT-tensors for n = 3, which provide Cauchy data at conformally flat null infinity . Specifically, this class of data is characterized for generating Λ > 0-vacuum spacetimes with two-symmetries, one of which axial, admitting a conformally flat . The class of data is infinite dimensional, depending on two arbitrary functions of one variable as well as a number of constants. Moreover, it contains the data for the Kerr–de Sitter spacetime, which we explicitly identify within.

我们展示了正交、未来有向基族的存在，这些基允许根据最少数量的参数将( ) 的每个偏斜对称自同态转换为单个规范形式。这种规范形式由每对元素共享，通过正交洛伦兹变换而不同，即它定义了正交洛伦兹群O ⁺ (1, n ) 在其代数上的伴随作用下的轨道。使用这种形式，我们获得 的商拓扑。来自球体的共形杀伤矢量场 (CKVF) 和之间的已知关系, CKVFs 的规范形式紧随其后。此形式用于查找同时涵盖所有情况的任意 CKVF 的自适应坐标。我们对偶数n进行计算，结果得到奇数n的情况。最后，我们使用适应的坐标来获得n = 3的宽类 TT 张量，它提供了共形平坦零无穷大的柯西数据。具体来说，这类数据的特征是生成 Λ > 0 真空时空，具有两个对称性，其中一个对称性为轴向，允许共形平坦. 数据类别是无限维的，取决于一个变量的两个任意函数以及许多常数。此外，它包含克尔-德西特时空的数据，我们在其中明确标识。