Abstract:Let $M^n$ be a compact orientable smooth Riemannian submanifold of dimension $n\geq 3$ in $\mathbb R^d$. We construct a family of deformed Hodge Laplacians $\Delta_t^*$, $t>0$, acting on differential forms and defined through the extrinsic geometry of $M^n$. We prove that these operators converge uniformly, in the appropriate operator topology, to the classical Hodge Laplacian $\Delta^*$ as $t\to0^+$. Given a point cloud $S_m \subset M^n$, we define empirical operators $\Delta^*_{t, S_m}$ and establish their spectral convergence in probability to $\Delta^*$, as $t \to 0^+$, under a suitable scaling regime $t = m ^{-\frac{1}{2n}}$. This rigorously extends the scalar Belkin--Niyogi Laplacian Eigenmaps framework to differential forms. As applications, we obtain consistent recovery procedures for the de Rham cohomology ring $H^* (M^n,\mathbf R)$, the second fundamental form of $M^n$, hence for the Riemannian curvature tensor, and consequently for the Pontryagin characteristic classes and Pontryagin numbers of $M^n$ from sampled data.
| Comments: | Revised version, condition $ n \ge 3$ added. 68 p |
| Subjects: | Differential Geometry (math.DG); Algebraic Topology (math.AT); Probability (math.PR); Statistics Theory (math.ST) |
| MSC classes: | 62R40, 58A12, 55N10, 53C40 |
| Cite as: | arXiv:2605.22265 [math.DG] |
| (or arXiv:2605.22265v2 [math.DG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.22265 arXiv-issued DOI via DataCite |
From: HongVan Le [view email]
[v1]
Thu, 21 May 2026 10:09:14 UTC (51 KB)
[v2]
Mon, 25 May 2026 15:33:23 UTC (55 KB)
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