发表机构
University of Notre Dame; Australian National University（圣母大学; 澳大利亚国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文定义了Reeb变换，并证明其在三维紧致曲面上具有单射性，可唯一重建曲面，但在更高维度中失效。
AI 中文摘要
Reeb变换提供了沿不同方向切片时$\nmathbb{R}^n$中形状如何变化的紧凑表示。我们将Reeb变换定义为由单位球面上所有方向的高度函数所诱导的Reeb图族。在本文中，我们对o-minimal可定义集上的Reeb变换进行了严格的处理，特别关注嵌入在$\nmathbb{R}^d$中的一维分层空间以及$\nmathbb{R}^3$中的曲面。我们在多种设置下建立了Reeb变换的单射性，包括$\nmathbb{R}^3$中的紧致曲面，捕获了唯一重建此类曲面所需的基本拓扑特征。然而，在三维以上的维度中，Reeb变换不再具有单射性，这表明该描述符在高维设置中的局限性。
英文摘要
Reeb Transforms offer a compact representation of how shapes in $\mathbb{R}^n$ change when sliced along varying directions. We define the Reeb Transform as the family of Reeb graphs induced by height functions along all directions in the unit sphere. In this paper, we develop a rigorous treatment of Reeb Transforms for o-minimal definable sets, with particular emphasis on one-dimensional stratified spaces embedded in $\mathbb{R}^d$ and on surfaces in $\mathbb{R}^3$. We establish injectivity of the Reeb Transform in multiple settings, including compact surfaces in $\mathbb{R}^3$, capturing the essential topological features needed to uniquely reconstruct such surfaces. However, in dimensions above three, the Reeb Transform ceases to be injective, indicating the limitations of this descriptor in higher-dimensional settings.
