Obviously Synge's function can be defined also in Riemannian manifolds and in that case it has non-negative sign. Generally speaking, Synge’s function is only locally defined and an attempt to define an extension to domains larger than convex normal neighborhoods generally leads to a multivalued function since there may be several geodesic segments joining a pair of points in the spacetime. It is however possible to define it in a neighborhood of the diagonal of M × × M , though this definition requires some arbitrary choice. Synge's world function (also its extension to a neighborhood of the diagonal of M × × M ) appears in particular in a number of theoretical constructions of quantum field theory in curved spacetime. It is the crucial object used to construct a parametrix of Green’s functions of Lorentzian Green hyperbolic 2nd order partial differential equations in a globally hyperbolic manifold, and in the definition of Hadamard Gaussian states.