Abstract We explore the geometric implications of introducing a spectral cut-off on compact Riemannian manifolds. This is naturally phrased in the framework of non-commutative geometry, where we work with spectral triples that are truncated by spectral projections of Dirac-type operators. We associate a metric space of ‘localized’ states to each truncation. The Gromov–Hausdorff limit of these spaces is then shown to equal the underlying manifold one started with. This leads us to propose a computational algorithm that allows us to approximate these metric spaces from the finite-dimensional truncated spectral data. We subsequently develop a technique for embedding the resulting metric graphs in Euclidean space to asymptotically recover an isometric embedding of the limit. We test these algorithms on the truncation of sphere and a recently investigated perturbation thereof.

中文翻译：

---

从谱三元组的截断重建流形

摘要 我们探讨了在紧凑黎曼流形上引入谱截止的几何含义。这自然是在非交换几何的框架中表述的，在那里我们使用被狄拉克型算子的谱投影截断的谱三元组。我们将“本地化”状态的度量空间与每个截断相关联。然后显示这些空间的 Gromov-Hausdorff 极限等于开始时的底层流形。这导致我们提出一种计算算法，该算法允许我们从有限维截断的光谱数据中近似这些度量空间。我们随后开发了一种技术，用于在欧几里德空间中嵌入得到的度量图，以渐近地恢复极限的等距嵌入。