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We consider the problem of the symmetry of the off-diagonal heat-kernelcoefficients as well as the coefficients corresponding to theshort-distance-divergent part of the Hadamard expansion in general smooth(analytic or not) manifolds. The requirement of such a symmetry played acentral r\^{o}le in the theory of the point-splitting one-loop renormalizationof the stress tensor in either Riemannian or Lorentzian manifolds. Actually,the symmetry of these coefficients has been assumed as a hypothesis in severalpapers concerning these issues without an explicit proof. The difficulty of adirect proof is related to the fact that the considered off-diagonalheat-kernel expansion, also in the Riemannian case, in principle, may be not aproper asymptotic expansion. On the other hand, direct computations of theoff-diagonal heat-kernel coefficients are impossibly difficult in nontrivialcases and thus no case is known in the literature where the symmetry does nothold. By approximating $C^\infty$ metrics with analytic metrics in common(totally normal) geodesically convex neighborhoods, it is rigorously proventhat, in general $C^\infty$ Riemannian manifolds, any point admits ageodesically convex neighborhood where the off-diagonal heat-kernelcoefficients, as well as the relevant Hadamard's expansion coefficients, aresymmetric functions of the two arguments.Comment: 30 pages, latex, no figures, minor errors corrected, English  improved, shortened version accepted for publication in Commun. Math. Phy

Proof of the Symmetry of the Off-Diagonal Heat-Kernel and Hadamard's Expansion Coefficients in General C ∞ Riemannian Manifolds