论文标题
$ \ mathbb {m}^{1，n} $中的偏斜成符号内态性：一种统一的规范形式
Skew-symmetric endomorphisms in $\mathbb{M}^{1,n}$: A unified canonical form with applications to conformal geometry
论文作者
论文摘要
我们展示了正直的，未来的指向基础的存在，这些家族允许$ \ mathbb {m}^{1，n} $（$ \ mathrm {skeWend}（\ m athbb {\ mathbb {m}^{1，n}）$）在单个canson can nibe上，将每一个分钟的依据送给$ \ mathbb {m}^{1，n} $（$ \ mathrm {skewend} $（$ \ mathrm {skewend}）（$ \ mathrm {skewend} $）的所有偏斜的代表性内态的存在。 $ \ mathrm {skewend}（\ mathbb {m}^{1，n}）中的每一元素共享这种规范的形式，其正向lorentz的正面转换与正面Lorentz组的Orbits在AldeDocter on Algebra的Algeoint actuction下不同。使用此表格，我们获得了$ \ mathrm {skewend}的商拓扑（\ mathbb {m}^{1，n}）/o^+（1，n）$。从$ \ mathrm {skewend}（\ mathbb {m}^{1，n}）$与球体的共形杀死矢量域（ckvfs）之间的已知关系中。该表格用于找到适用于任意CKVF的适应坐标，该坐标同时涵盖了所有情况。我们对$ n $进行计算，并因此获得奇数$ n $的情况。最后，我们采用改编的坐标来获得$ n = 3 $的宽类TT量，这些TT量在同轴纯净的Infinity $ \ Mathscr {i} $上提供了Cauchy数据。具体而言，此类数据的特征是生成$λ> 0 $ -Vacuum spaceTimes具有两个符号的时间，其中一个轴向轴向承认了一个合成平坦的$ \ mathscr {i} $。数据类是无限的维度，具体取决于一个变量和许多常数的两个任意函数。此外，它包含了Kerr-De保姆时空的数据，我们在其中明确识别。
We show the existence of families of orthonormal, future directed bases which allow to cast every skew-symmetric endomorphism of $\mathbb{M}^{1,n}$ ($\mathrm{SkewEnd}(\mathbb{M}^{1,n})$) in a single canonical form depending on a minimal number of parameters. This canonical form is shared by every pair of elements in $\mathrm{SkewEnd}(\mathbb{M}^{1,n})$ differing by an orthochronous Lorentz transformation, i.e. it defines the orbits of the orthochronous Lorentz group under the adjoint action on its algebra. Using this form, we obtain the quotient topology of $\mathrm{SkewEnd}(\mathbb{M}^{1,n})/O^+(1,n)$. From known relations between $\mathrm{SkewEnd}(\mathbb{M}^{1,n})$ and the conformal Killing vector fields (CKVFs) of the sphere $\mathbb{S}^n$, 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 $\mathscr{I}$. Specifically, this class of data is characterized for generating $Λ>0$-vacuum spacetimes with two-symmetries, one of which axial, admitting a conformally flat $\mathscr{I}$. 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.
