Which is my point - that is the error you make as I explained in the part of my post you did not quote:

**A Riemann manifold is not a curved hyperplane**. You do not have any Riemann manifold in your theory. You do not have any S

_{g} in a Riemann manifold. You also have S

_{g} = 0 (or undefined).

This is a

hyperplane. Note that it is embedded in an ambient space. Curvature may only be defined by that ambient space

This is a

Pseudo-Riemannian manifold as used in GR. Note that there is no mention of an ambient space. That is why in the Einstein Field equations, there are indexes that go 1, 2, 3 ,4 (3 space dimensions and a time dimension) but not 5 for an ambient space. The curvature in GR is intrinsic and does not refer to any ambient space.