I would like to get some foundational concepts straight about GR as it is currently understood (I guess it is not seen exactly the same it was almost a century ago even if the basic concepts remain, this I would like to elucidate here too). For instance, I understand that a basic axiom of GR as it was presented by Einstein in 1915 is the existence of a curved Lorentzian 4-manifold that is taken as the starting point of the theory in which the EFE are applied. Is this generally agreed as a basic postulate of GR? It would seem Einstein's idea of gravity as curvature demands such a postulate. My bringing this up arises from considering the always problematic concepts of background independence and gauge invariance in GR. According to this more modern view of GR usually put forward by QG theoriticians the metric of the manifold (as implied in the "curved Lorentzian manifold axiom above mentioned) seems to lose its axiomatic importance in favor of the connection and the existence of a smooth manifold as basic postulate, relegating metrics to particular solutions with particular physical (idealized or not) circumstances. So the doubt comes to mind what the most basic axiom of GR would be, the existence of a smooth manifold with a certain connection or the old gravity is curvature that demands the existence of a Lorentzian metric with curvature? Or both at the same time?