A Local Lorentz invariance of propagator on curved spaces

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I am looking for a proof that the Feynman propagator is locally a lorentz invariant (at least for scalar fields) also in curved space-times if the background geometry is smooth enough.
I mean, since it is of course a lorentz invariant on flat spaces, this should also be true if a choose a sufficiently small portion of the space and the background geometry is sufficiently smooth, even if it should contain non-local terms when evaluated between well distanced spacetime points.
Some advice on where I can find something like that?
 
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