AI 中文总结
研究二维球面共形几何，通过特定对称无迹张量导出线性二阶偏微分方程，其解空间有闵可夫斯基结构，存在无穷多相关空间，且二维球面能规范等距嵌入闵可夫斯基空间。
AI 中文摘要
我们讨论二维球面的几何。我们的视角聚焦于将给定度量重标为圆球度量的共形因子集。我们发现球面上存在一个唯一的对称且无迹张量，它与里奇张量互补成一个无散对称张量。利用此张量，我们导出一个线性二阶偏微分方程，其解空间带有自然的闵可夫斯基结构。事实上存在无穷多个这样的空间，它们都通过等距变换相关联，这些等距变换可由‘4 - 向量’参数化。最后，我们表明每个二维球面都能以一种规范方式等距嵌入到闵可夫斯基空间中。
英文摘要
We discuss the geometry of the two-dimensional sphere. Our perspective focuses on the set of conformal factors that rescale a given metric to a round metric. We find that there exists a unique symmetric and trace-free tensor on the sphere which complements the Ricci tensor to a divergence-free symmetric tensor. Using this tensor, we derive a linear second-order PDE whose solution space carries a natural Minkowski structure. We show that there are in fact infinitely many such spaces, all related by isometries which can be thought of as being parametrised by ``4-vectors''. Finally, we show that every 2-sphere can be isometrically embedded in a Minkowski space in a canonical way.
Comments12 pages
