Reflecting brownian motion and the gauss-bonnet-chern theorem

TL;DR

This paper employs reflecting Brownian motion to provide a probabilistic proof of the Gauss-Bonnet-Chern theorem for compact Riemannian manifolds with boundary, focusing on boundary local time analysis.

Contribution

It introduces a novel probabilistic approach using RBM to prove the Gauss-Bonnet-Chern theorem, emphasizing boundary behavior.

Findings

Boundary integrand derived from local time asymptotics

Probabilistic proof complements classical differential geometry methods

Analysis of boundary local time for small times

Abstract

We use reflecting Brownian motion (RBM) to prove the well known Gauss-Bonnet-Chern theorem for a compact Riemannian manifold with boundary. The boundary integrand is obtained by carefully analyzing the asymptotic behavior of the boundary local time of RBM for small times.