Elementary proof of symmetry of the off-diagonal Seeley-DeWitt (and related Hadamard) coefficients

TL;DR

This paper provides an elementary proof demonstrating the symmetry of off-diagonal Seeley-DeWitt and Hadamard coefficients on smooth manifolds of any signature, simplifying understanding of their properties.

Contribution

It introduces a straightforward proof of symmetry for these coefficients, extending their known properties to arbitrary signature manifolds.

Findings

Off-diagonal Seeley-DeWitt coefficients are symmetric.

Off-diagonal Hadamard coefficients are symmetric.

The proof applies to manifolds of any signature.

Abstract

We will prove in an elementary way that off-diagonal Seeley-DeWitt and Hadamard coefficients are (sesqui-)symmetric for smooth manifolds of arbitrary signature.