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

Abstract

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 rôle 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.

arXiv comment30 pages, latex, no figures, minor errors corrected, English improved, shortened version accepted for publication in Commun. Math. Phys

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