AI 中文总结
该研究证明伪黎曼子流形的普通、类空、类时接触数等价且重合，构造了接触数均为6的洛伦兹曲面，将无穷接触极限纳入半黎曼理论。
AI 中文摘要
设$M_s^n$为伪欧几里得空间中具有不定诱导度量的非退化伪黎曼子流形。将接触条件限制在类空或类时切方向上，会得到两个先验不同的单侧接触数。我们证明，对于任意给定阶数，普通接触、类空接触和类时接触是等价的，因此三个接触数重合。证明采用归纳法：单侧接触假设首先给出第二基本形式的迭代共变导数的高阶正交恒等式；随后通过多项式论证，将这些恒等式的对角标量表达式从任一单位伪球面扩展到整个切空间；再通过第二次归纳证明因果不变性。我们还在$\boldsymbol{\rm E}_4^8$中构造了一个显式洛伦兹曲面，其普通、类空和类时接触数均等于6，并将无穷接触极限置于测地法截面与螺旋浸入的半黎曼理论中。
英文摘要
Let $M_s^n$ be a non-degenerate pseudo-Riemannian submanifold of a pseudo-Euclidean space with indefinite induced metric. Restricting the contact condition to spacelike or to timelike tangent directions gives two a priori different one-sided contact numbers. We prove that, for every prescribed order, ordinary, spacelike and timelike contact are equivalent; consequently, the three contact numbers coincide. The proof is inductive. A one-sided contact hypothesis first yields higher-order orthogonality identities for the iterated covariant derivatives of the second fundamental form. Their diagonal scalar expressions are then extended from either unit pseudo-sphere to the whole tangent space by a polynomial argument, and a second induction proves causal invariance. We also construct an explicit Lorentzian surface in $\mathbb E_4^8$ whose ordinary, spacelike and timelike contact numbers are all equal to six, and place the infinite-contact limit in the semi-Riemannian theory of geodesic normal sections and helical immersions.
