论文标题
曲折道路理论
Twistor Theory of Dancing Paths
论文作者
论文摘要
给定表面$ \ MATHCAL {U} $上的路径几何形状，我们在四个manifold上构建了一个因果结构，该结构是$ \ MATHCAL {U} $的非属性对（点，路径）的配置空间。当且仅当$ \ Mathcal {u} $是一个真实的投影平面时，该因果结构对应于共形结构，而路径是行。我们举例说明了对称六元的因果结构的示例，该结构在$ {\ rm sl}（2，{\ mathbb r}）上对应于$ {\ mathbb r}）$ - 不变的投影结构，其中路径是以原点为中心的面积$π$的椭圆形。我们还将在七维流形上讨论与射击平面上非事件对（点，圆锥）相对应的七维歧管上的因果结构。
Given a path geometry on a surface $\mathcal{U}$, we construct a causal structure on a four-manifold which is the configuration space of non-incident pairs (point, path) on $\mathcal{U}$. This causal structure corresponds to a conformal structure if and only if $\mathcal{U}$ is a real projective plane, and the paths are lines. We give the example of the causal structure given by a symmetric sextic, which corresponds on an ${\rm SL}(2,{\mathbb R})$-invariant projective structure where the paths are ellipses of area $π$ centred at the origin. We shall also discuss a causal structure on a seven-dimensional manifold corresponding to non-incident pairs (point, conic) on a projective plane.
