Empirical Hodge Laplacians, Cohomology Ring, and Manifold Learning

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.