The Intrinsic Geometry of Some Random Manifolds

TL;DR

This paper investigates the almost sure convergence of random manifold embeddings into high-dimensional Euclidean spaces, demonstrating that the limits are deterministic and analyzing the convergence of intrinsic geometric functionals like Lipschitz-Killing curvatures.

Contribution

It establishes the almost sure convergence of random manifold embeddings and shows that intrinsic functionals converge to deterministic limits, including unbiasedness results for Lipschitz-Killing curvatures.

Findings

Random embeddings converge almost surely to deterministic limits

Intrinsic functionals of embedded manifolds also converge deterministically

Lipschitz-Killing curvatures are unbiased and have explicit limits

Abstract

We study the a.s. convergence of a sequence of random embeddings of a fixed manifold into Euclidean spaces of increasing dimensions. We show that the limit is deterministic. As a consequence, we show that many intrinsic functionals of the embedded manifolds also converge to deterministic limits. Particularly interesting examples of these functionals are given by the Lipschitz-Killing curvatures, for which we also prove unbiasedness, using the Gaussian kinematic formula.