A stochastic Gauss-Bonnet-Chern formula

TL;DR

This paper establishes a probabilistic version of the Gauss-Bonnet-Chern theorem by linking Gaussian random sections of vector bundles to geometric invariants like the Euler form, revealing new connections between probability and differential geometry.

Contribution

It introduces a canonical metric and compatible connection derived from Gaussian ensembles of random sections and proves a refined probabilistic Gauss-Bonnet-Chern theorem relating expected zero-loci to the Euler form.

Findings

Gaussian ensembles induce canonical metrics and connections on vector bundles

Expected zero-locus currents match the Euler form for oriented bundles and manifolds

Probabilistic interpretation of classical topological invariants

Abstract

We prove that a Gaussian ensemble of smooth random sections of a real vector bundle over compact manifold canonically defines a metric on the bundle together with a connection compatible with it. Additionally, we prove a refined Gauss-Bonnet-Chern theorem stating that if the bundle and the manifold are oriented, then the Euler form of the above connection can be identified, as a current, with the expectation of the random current defined by the zero-locus of a random section in the above Gaussian ensemble.