Invariants of the heat equation
type: Item
instance of: scholarly article
title: Invariants of the heat equation
zbMATH Open document ID: 0584.58041
published in: Pacific Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 3936048
zbMATH Keywords: invariant
zbMATH Keywords: self-adjoint elliptic differential operator
zbMATH Keywords: Atiyah-Singer index theorem
zbMATH Keywords: Gauss-Bonnet theorem
zbMATH Keywords: zeta and eta functions
author: Howard D. Fegan
author: Peter B. Gilkey
MaRDI profile type: Publication
full work available at URL: pjm.1985.117.233
OpenAlex ID: W2004623930
DOI: 10.2140/PJM.1985.117.233
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description: scientific article; zbMATH DE number 3936048