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

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.