Heat kernel expansions in vector bundles

Let MM be a complete connected smooth (compact) Riemannian manifold of dimension nn. Let Π:V→MΠ:V→M be a smooth vector bundle over MM. Let L=12Δ+b be a second order differential operator on MM, where ΔΔ is a Laplace-Type operator on the sections of the vector bundle VV and bb a smooth vector field on MM. Let kt(−,−)kt(−,−) be the heat kernel of VV relative to LL. In this paper we will derive an exact and an asymptotic expansion for kt(x,y0)kt(x,y0) where y0y0 is the center of normal coordinates defined on MM, xx is a point in the normal neighborhood centered at y0y0. The leading coefficients of the expansion are then computed at x=y0x=y0 in terms of the linear and quadratic Riemannian curvature invariants of the Riemannian manifold MM, of the vector bundle VV, and of the vector bundle section ϕϕ and its derivatives.We end by comparing our results with those of previous authors (I. Avramidi, P. Gilkey, and McKean–Singer).