A probabilistic proof of the Gauss-Bonnet formula for manifolds with boundary

TL;DR

This paper presents a probabilistic proof of the Gauss-Bonnet formula for compact Riemannian manifolds with boundary, utilizing stochastic analysis and the Feynman-Kac formula to connect geometry and topology.

Contribution

It introduces a novel probabilistic approach to the Gauss-Bonnet formula for manifolds with boundary, extending previous methods to include boundary effects.

Findings

Path integral representation of Euler characteristic using reflected Brownian motion

Clarification of boundary contributions via shape operator analysis

Derivation of local Gauss-Bonnet formula leading to global result

Abstract

In this short note we outline a simple probabilistic proof of the Gauss-Bonnet formula for compact Riemannian manifolds with boundary, which adapts to this setting an argument due to Hsu \cite{Hs1,Hs2} in the closed case. The new technical ingredient is the Feynman-Kac formula for differential forms satisfying absolute boundary conditions proved in \cite{dL}. Combined with the so-called supersymmetric aproach to index theory, this leads to a path integral representation of the Euler characteristic of the manifold in terms of normally reflected Brownian motion whose short time asymptotics clarifies the role played by the shape operator in determining the boundary contribution to the formula. As a consequence we obtain the expected {\em local} Gauss-Bonnet formula which upon integration yields the desired global result.