论文标题
向量字段对音量有什么了解？
What does a vector field know about volume?
论文作者
论文摘要
本说明为Viterbo的问题提供了一个肯定的答案，该问题是关于共享相同Reeb vector字段的非构象联系表的存在。从Croke-Kleiner和Abbondanlo的观察开始，即这种联系形式定义了相同的总体积，我们讨论了更广泛的Geodessible Vector Fields类别的各种相关问题。特别是，我们定义了相关的基本共同体中的欧拉类别的欧拉类，并给出了消失的欧拉类别的向量场的拓扑表征。我们使用截面的全局表面以及由地理位置可见的矢量场确定的体积，证明了高斯 - 摩托克和庞加尔 - 霍普的定理。该体积是针对Seifert Fibred 3 manifolds和一些横向溶性流量计算的。
This note provides an affirmative answer to a question of Viterbo concerning the existence of nondiffeomorphic contact forms that share the same Reeb vector field. Starting from an observation by Croke-Kleiner and Abbondandolo that such contact forms define the same total volume, we discuss various related issues for the wider class of geodesible vector fields. In particular, we define an Euler class of a geodesible vector field in the associated basic cohomology and give a topological characterisation of vector fields with vanishing Euler class. We prove the theorems of Gauss-Bonnet and Poincaré-Hopf for closed, oriented 2-dimensional orbifolds using global surfaces of section and the volume determined by a geodesible vector field. This volume is computed for Seifert fibred 3-manifolds and for some transversely holomorphic flows.
