Proof of the symmetry of the off-diagonal heat-kernel and Hadamard's expansion coefficients in general $C^{\infty}$ Riemannian manifolds

Details Proof of the symmetry of the off-diagonal heat-kernel and Hadamard's expansion coefficients in general $C^{\infty}$ Riemannian manifolds Proof of the symmetry of the off-diagonal heat-kernel and Hadamard's expansion coefficients in general $C^{\infty}$ Riemannian manifolds

1999-02-11

arxiv.org/abs/gr-qc/9902034v2

Commun.Math.Phys. 208 (1999) 283-309

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

30 pages, latex, no figures, minor errors corrected, English improved, shortened version accepted for publication in Commun. M

Scientific paper

10.1007/s002200050759

We consider the problem of the symmetry of the off-diagonal heat-kernel coefficients as well as the coefficients corresponding to the short-distance-divergent part of the Hadamard expansion in general smooth (analytic or not) manifolds. The requirement of such a symmetry played a central r\^{o}le in the theory of the point-splitting one-loop renormalization of the stress tensor in either Riemannian or Lorentzian manifolds. Actually, the symmetry of these coefficients has been assumed as a hypothesis in several papers concerning these issues without an explicit proof. The difficulty of a direct proof is related to the fact that the considered off-diagonal heat-kernel expansion, also in the Riemannian case, in principle, may be not a proper asymptotic expansion. On the other hand, direct computations of the off-diagonal heat-kernel coefficients are impossibly difficult in nontrivial cases and thus no case is known in the literature where the symmetry does not hold. By approximating $C^\infty$ metrics with analytic metrics in common (totally normal) geodesically convex neighborhoods, it is rigorously proven that, in general $C^\infty$ Riemannian manifolds, any point admits a geodesically convex neighborhood where the off-diagonal heat-kernel coefficients, as well as the relevant Hadamard's expansion coefficients, are symmetric functions of the two arguments.

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