Singularities of flat fronts in hyperbolic space
Singularities of flat fronts in hyperbolic space
复制标题
DOI：
10.2140/pjm.2005.221.303
复制
发表时间：
2005-10-01
影响因子：
0.6
通讯作者：
Yamada, K
中科院分区：
文献类型：
--
作者：
Kokubu, M;Rossman, W;Yamada, K
众所周知，任何黎曼流形的单位余切丛都具有正则接触结构。如果黎曼 3 流形中的表面是勒曼浸入到单位余切丛中的投影，则该表面称为前沿。我们给出了易于计算的标准，将正面上的奇点定义为尖边或燕尾。利用这一点，我们证明了双曲 3 空间中的一般平坦前沿仅允许尖端边缘和燕尾。我们还表明，任何完整的平坦正面（假设它不是旋转对称的）都具有相关的平行表面，其奇点仅由尖端边缘和燕尾组成。
It is well-known that the unit cotangent bundle of any Riemannian manifold has a canonical contact structure. A surface in a Riemannian 3-manifold is called a front if it is the projection of a Legendrian immersion into the unit cotangent bundle. We give easily computable criteria for a singular point on a front to be a cuspidal edge or a swallowtail. Using this, we prove that generically flat fronts in hyperbolic 3-space admit only cuspidal edges and swallowtails. We also show that any complete flat front (provided it is not rotationally symmetric) has associated parallel surfaces whose singularities consist of only cuspidal edges and swallowtails.