Reflecting Brownian motion and the Gauss-Bonnet-Chern theorem (Q6169185)

This paper treats the well-known Gauss-Bonnet-Chern theorem for a compact manifold with boundary. The authors adopt a probabilistic approach to use reflecting Brownian motion (RBM) in order to prove the theorem. One of the peculiar features consists in the fact that the boundary integrand is obtained by carefully analyzing the asymptotic behavior of the boundary local time of RBM for small times. More precisely, let \(M\) be a smooth oriented manifold with boundary, and let \(\chi (M)\) be its Euler characteristic defined by \[ \chi (M) = \sum_{i=0}^m (-1)^i b_i, \] where \(b_i = \dim H^i(M)\) are the Betti numbers, and \(H^i(M)\) is the \(i\)-dimensional cohomology group. Assume that the manifold \(M\) is equipped with a Riemannian metric. A fundamental result in differential geometry is the Gauss-Bonnet-Chern (GBC) theorem, cf. [\textit{S.-S. Chern}, Ann. Math. (2) 45, 747--752 (1944; Zbl 0060.38103)] for a simple proof of the GBC theorem for a closed Riemannian manifolds. The theorem expresses the Euler characteristic as the sum of two integrals on \(M\) and on its boundary \(\partial M\) \[ \chi(M) = \int_M e_M(x) dx + \int_{ \partial M} e_{\partial M} ( \bar{x} ) d \bar{x}, \] where \(e_M\) (resp. \(e_{ \partial M}\) ) is a local geometric invariant determined by the curvature tensor (resp. the second fundamental form of the boundary of the manifold) respectively. For a manifold without boundary, \textit{H. P. McKean jun.} and \textit{I. M. Singer} [J. Differ. Geom. 1, 43--69 (1967; Zbl 0198.44301)] expressed the integral of the Euler form on the manifold in terms of the supertrace \(str\) of the heat kernal \(p^*\) of the Hodge-de Rham Laplacian on the bundle of differential forms: \[ \chi(M) = \int_M str \,\,\, p^* (t,x,x ) dx. \] On the other hand, the probabilistic approach to analytic index theorems was initiated by \textit{J.-M. Bismut} [in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 491--504 (1987; Zbl 0693.58023)] for Riemannian manifolds without boundary. Roughly speaking, the above approach is based upon the fact that the heat kernel on differential forms associated with the Hodge-de Rham Laplacian of the horizontal Brownian motion on the frame bundle of the Riemannian manifold. However, Bismut's approach cannot be easily carried over to a manifold with boundary. The major difficulty lies in the fact that the multiplicative functional is discontinuous at the boundary. Another difficulty is the presence of the boundary local time, with the result that the algebraic ``fantastic cancellation (FC)'' does not occur at the path level, although FC works well in the Bismut's work. Under these circumstances, using Malliavin calculus, \textit{I. Shigekawa} et al. [Osaka J. Math. 26, No. 4, 897--930 (1989; Zbl 0704.58056)] successfully handled the singular terms on the boundary, thus giving a probabilistic proof of the Gauss-Bonnet-Chern formula. Malliavin calculus allows the heat kernel to be regarded as a generalized functional on Brownian motion which can then be expanded as an asymptotic power series in the small time parameter in a generalized sense. This probabilistic approach cannot be regarded as a direct extension of the Bismut approach because of the use of Malliavin calculus. The purpose of the present work is to give an elementary probabilistic proof of the Gauss-Bonnet-Chern theorem without using Malliavin calculus. Their approach is heavily indebted to the previous works by \textit{E. P. Hsu} [Stochastic analysis on manifolds. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 0994.58019)] and Shigekawa et al. [loc. cit.].