Reflecting Brownian Motion and the Gauss–Bonnet–Chern Theorem

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.

Keywords

Manifold with boundary / Gauss–Bonnet–Chern theorem / Reflecting Brownian motion