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 q _t (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 θ _P . 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.