Relating the second and fourth order terms in the asymptotic expansion of the heat equation (Q678185)

Let \(D\) be an operator of Laplace type on the space of sections to a smooth line bundle \(L\) over a compact smooth Riemannian manifold \(M\) of dimension \(m\) without boundary. As \(t\downarrow 0\), there is an asymptotic expansion \[ \text{Tr}_{L^2}e^{-tD}\approx\Sigma_{n\geq0}a_n(D)t^{(n-m)/2}; \] \(a_n\) is given by a local formula: \(a_n(D)=\int_Ma_n(x,D)dx\). In the context of both Riemannian and affine geometry, the authors study inequalities between the invariants \(a_4(D)\) and \(C(m)\int_Ma_2^2(x,D)dx\) where \(C(m)\) is a suitably chosen normalizing constant.