Global smooth fibrations of ℝ3 by oriented lines
Global smooth fibrations of ℝ3 by oriented lines
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ℝ3 通过定向线的全局平滑纤维
DOI：
10.1112/blms/bdn115
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发表时间：
2009
影响因子：
0.9
通讯作者：
M. Salvai
中科院分区：
文献类型：
--
作者：
M. Salvai
ℝ3 通过定向线的平滑纤维化由 ℝ3 上的平滑单位矢量场 V 给出，其所有积分曲线都是直线。如果 dV 仅在 V 方向上消失，则这种纤维化被称为非简并。令 ℒ 为 ℝ3 的定向线空间，赋予其规范的伪黎曼中性度量。我们通过定向线将ℝ3的非简并光滑纤维表征为ℒ中的闭合（在相对拓扑中）确定的连通表面。特别是，ℒ 上的局部条件意味着存在全局纤维化。此外，对于任何此类纤维化，基底空间与开放圆盘是微分同胚的，并且纤维的方向形成双球体的开放凸集。我们也以类似的方式表征平滑（可能是简并）纤维。
A smooth fibration of ℝ3 by oriented lines is given by a smooth unit vector field V on ℝ3 all of whose integral curves are straight lines. Such a fibration is said to be nondegenerate if dV vanishes only in the direction of V. Let ℒ be the space of oriented lines of ℝ3 endowed with its canonical pseudo‐Riemannian neutral metric. We characterize the nondegenerate smooth fibrations of ℝ3 by oriented lines as the closed (in the relative topology) definite connected surfaces in ℒ. In particular, local conditions on ℒ imply the existence of a global fibration. Besides, for any such fibration the base space is diffeomorphic to the open disc and the directions of the fibers form an open convex set of the two‐sphere. We characterize as well, in a similar way, the smooth (possibly degenerate) fibrations.