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We consider the global Hadamard condition and the notion of Hadamardparametrix whose use is pervasive in algebraic QFT in curved spacetime (seerefences in the main text). We point out the existence of a technical problemin the literature concerning well-definedness of the global Hadamard parametrixin normal neighbourhoods of Cauchy surfaces. We discuss in particular thedefinition of the (signed) geodesic distance $\sigma$ and related structures inan open neighbourhood of the diagonal of $M\times M$ larger than $U\times U$,for a normal convex neighborhood $U$, where $(M,g)$ is a Riemannian orLorentzian (smooth Hausdorff paracompact) manifold. We eventually propose aquite natural solution which slightly changes the original definition by B.S.Kay and R.M. Wald and relies upon some non-trivial consequences of theparacompactness property. The proposed re-formulation is in agreement with M.J.Radzikowski's microlocal version of the Hadamard condition.

On the global Hadamard parametrix in QFT and the signed squared geodesic distance defined in domains larger than convex normal neighbourhoods