Leading terms in the heat invariants

Let $D$ be a second-order differential operator with leading symbol given by the metric tensor on a compact Riemannian manifold. The asymptotics of the heat kernel based on $D$ are given by homogeneous, invariant, local formulas. Within the set of allowable expressions of a given homogeneity there is a filtration by degree, in which elements of the smallest class have the highest degree. Modulo quadratic terms, the linear terms integrate to zero, and thus do not contribute to the asymptotics of the ${L^2}$ trace of the heat operator; that is, to the asymptotics of the spectrum. We give relations between the linear and quadratic terms, and use these to compute the heat invariants modulo cubic terms. In the case of the scalar Laplacian, qualitative aspects of this formula have been crucial in the work of Osgood, Phillips, and Sarnak and of Brooks, Chang, Perry, and Yang on compactness problems for isospectral sets of metrics modulo gauge equivalence in dimensions 2 and 3.