  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.
