The Gauss-Bonnet-Chern theorem: a probabilistic perspective

TL;DR

This paper offers a probabilistic interpretation of the Gauss-Bonnet-Chern theorem by linking the Euler form to the expected zero-locus of random sections, and shows how to recover geometric data from random section statistics.

Contribution

It introduces a novel probabilistic perspective on the Euler form and demonstrates how to reconstruct metric and connection information from random sections.

Findings

Euler form equals the expectation of the zero-locus current of a random section

Method to reconstruct metric and connection from random section statistics

Probabilistic interpretation of classical topological invariants

Abstract

We prove that the Euler form of a metric connection on real oriented vector bundle $E$ over a compact oriented manifold $M$ can be identified, as a current, with the expectation of the random current defined by the zero-locus of a certain random section of the bundle. We also explain how to reconstruct probabilistically the metric and the connection on $E$ from the statistics of random sections of $E$.