Brownian Motion and Riemannian Geometry in the Neighbourhood of a Submanifold

Abstract

Let M be a complete connected smooth Riemannian manifold of dimension n and P a q-dimensional smoothly embedded smooth submanifold of M. M0 will denote a tubular neighbourhood of P in M. Let L = \(\frac{1}{2}\Delta\) + b + c be a differential operator on M, where Δ is the Laplacian on smooth functions, b a smooth vector field on M and c a smooth potential term. Let p\(_{t}^{\mathrm{M}_{0}}(-,-)\) be the Dirichlet heat kernel of M0, and p\(_{t}^{\mathrm{M}}(-,-)\) the heat kernel of M. We will show in this article that for a smooth function f:M→R with compact support in M0, the integral \(\int_{\mathrm{P}}\)f(y)p\(_{t}^{\mathrm{M}_{0}}\)(x,y)π(dy) generalizes the usual Dirichlet heat kernel and has an asymptotic expansion of the form: $$ \int_{\mathrm{P}}f(y)p_{t}^{\mathrm{M}_{0}}(x,y)\pi({\rm dy}) = q_{t} (x,P)\left[ \mathrm{f}(\gamma(\mathrm{t}))+\sum\limits_{\alpha=1}^{N}\mathrm{b}_{\alpha}\mathrm{(x,P)t}^{\alpha}+ \mathrm{o}(t^{N})\right] , $$ where π is the Riemannian measure on P and qt (x,P) is defined in Eq. 2.7. The asymptotic expansion is then extended to \(\int_{\mathrm{P}}\)f(y)p\(_{t}^{\mathrm{M}}\)(x,y)π(dy). The above expansion generalizes the usual Minakshisundaram–Pleijel heat kernel expansion and a computation of the leading expansion coefficients suggests that it is also a generalization of the heat content expansion. The expansion coefficients are local geometric invariants given by simple integrals of the derivatives of the metric tensor and the volume change factor θP. The leading coefficients are then computed in terms of the Riemannian geometry in the neighbourhood of the submanifold P at the centre of Fermi coordinates y0 ∈ P.