论文标题
在平坦的3型manifolds上的地球和共同点Reeb矢量场
Geodesic and conformally Reeb vector fields on flat 3-manifolds
论文作者
论文摘要
如果Riemannian歧管$ M $上的单位矢量场被称为Geodesic，则如果其所有积分曲线都是大地测量曲线。如果$ m $是一个平坦的3个manifold，我们表明每个这样的矢量字段与二维完全的大地叶叶相切。此外，据表明，当且仅当且仅当有其他地球矢量范围的正交互补的触点结构中，且仅当触点结构横向与$ x $时，the flat 3 manifold上的地球矢量字段$ x $是接触表格的reeb矢量场（直至进行缩放）。根据$ x $的体积给出了对3多头的提升联系结构（达到差异）的明确描述。最后，讨论了非关闭平面3个体的类似结果。
A unit vector field on a Riemannian manifold $M$ is called geodesic if all of its integral curves are geodesics. We show, in the case of $M$ being a flat 3-manifold not equal to $\mathbb{E}^3$, that every such vector field is tangent to a 2-dimensional totally geodesic foliation. Furthermore, it is shown that a geodesic vector field $X$ on a closed flat 3-manifold is (up to rescaling) the Reeb vector field of a contact form if and only if there is a contact structure transverse to $X$ that is given as the orthogonal complement of some other geodesic vector field. An explicit description of the lifted contact structures (up to diffeomorphism) on the 3-torus is given in terms of the volume of $X$. Finally, similar results for non-closed flat 3-manifolds are discussed.
