The Earth is flat in curved spacetime.
It’s more interesting than it sounds since every point on the surface of the Earth is flat relative to everywhere in its vicinity. Otherwise a marble would roll towards the horizon. “The Earth is so big, you can’t see its curvature,” will be the obvious rejoinder. But who is to say that a view of the Earth from outer space is more accurate than a view from the Earth itself? Some things are more perceptible from a great distance, but others are less. Temperature, for instance, is impossible to perceive exempt in proximity. Flatness may be akin to temperature in this respect since we experience it qualitatively, through our sense of balance or equilibrium. In any case, I don’t think it is a foregone conclusion that “flatness” is more perceptible from a distance than from proximity.
The whole question is very confusing and the confusion arises in a large part because, as is so often the case, scientific discourse appropriates terms from common language and then alters their meanings. Ordinarily, “flat” describes a phenomenological observation that is corroborated primarily by the sense of sight and also the kinesthetic sense or the sense of balance. The question of what point of reference lends itself to the most truthful observations has already been posed. Again, I do not think that the perspective of outer space, or no perspective at all, is necessarily the correct one.
In the “view-from-nowhere” perspective of scientific discourse, “flat” cannot really have a referent for its ordinary usage because all of the qualitative aspects contained in its meaning are outside of scientific consideration. “Flat” refers to something that can be described by Euclidian geometry. To say “space is curved” means it requires alternate geometries to describe it. To say “spacetime is a manifold” means that local areas of space are “flat” but spacetime as a whole is actually “curved.” Again, this may be internally consistent but it is very far from what we understand by the word “flat” in ordinary parlance.
So the Earth is flat in curved spacetime.