Given a flow on a 3-dimensional integral homology sphere, we give a formula for the Euler characteristic of its transverse surfaces, in terms of boundary data only. We illustrate the formula with several examples, in particular with surfaces of low genus. As an application, we show that for a right-handed flow with an ergodic invariant measure, the genus is an asymptotic invariant of order 2 proportional to the helicity.

中文翻译：

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维度 3 中的向量场和属

给定一个 3 维积分同调球面上的流动，我们给出了其横向表面的欧拉特性的公式，仅根据边界数据。我们用几个例子来说明这个公式，特别是低属曲面。作为一个应用，我们证明了对于具有遍历不变测度的右手流，属是一个与螺旋度成正比的 2 阶渐近不变量。