Invariants of the heat equation (Q1069533)

Let M be a compact Riemannian manifold without boundary and \(P: C^{\infty}(V)\to C^{\infty}(V)\) be a self-adjoint elliptic differential operator with leading symbol p(x,\(\xi)\), a matrix for \(x\in M\) and \(\xi \in T^*M_ x\). The authors demonstrate that if p(x,\(\xi)\) is positive definite then the set \(\{e_ n(x,\exp (-tP))\}\), which are obtained by expanding the kernel K(x,x) of the operator exp(-tP) when \(t>0\), span all the invariants arising from the heat equation Tr exp(- tP). In particular they assert that no new invariant could be obtained from \(e_ n(x,A(P)\exp (-tB(P))\), where A(r) and B(r) are constant coefficient polynomials. In the case that p(x,\(\xi)\) is indefinite the set \(\{e_ n(x,\exp (-tP^ 2),e_ n(x,P \exp (-tP^ 2))\}\) span all invariants. The authors emphasize the importance of studying the invariant of heat equation by pointing out that this problem is related to a number of profound theorems such as the Atiyah-Singer index theorem, the Gauss-Bonnet theorem. The authors also show that the above-mentioned invariants are closely related to the study of zeta and eta functions.