Heat kernel expansions in vector bundles

Abstract

Let M be a complete connected smooth (compact) Riemannian manifold of dimension n . Let Π : V → M be a smooth vector bundle over M . Let L = 1 2 Δ + b be a second order differential operator on M , where Δ is a Laplace-Type operator on the sections of the vector bundle V and b a smooth vector field on M . Let k t ( − , − ) be the heat kernel of V relative to L . In this paper we will derive an exact and an asymptotic expansion for k t ( x , y 0 ) where y 0 is the center of normal coordinates defined on M , x is a point in the normal neighborhood centered at y 0 . The leading coefficients of the expansion are then computed at x = y 0 in terms of the linear and quadratic Riemannian curvature invariants of the Riemannian manifold M , of the vector bundle V , and of the vector bundle section ϕ and its derivatives. We end by comparing our results with those of previous authors (I. Avramidi, P. Gilkey, and McKean–Singer).