You're right about that first part. I hadn't considered the interface between ground and air. I retract that. Still, I'm skeptical about any model that's built up from only one kind of measurement or observation, or at least ones as complex as the ones in this area. I accept that the ground/air interface exists because there is more than one way to show its presence (to ignore the fact that it's just obvious), but there's only one way to observe the interfaces under the Earth's surface.

Not really.

There are other methods.

We know that the pressure will increase as you go down, as it needs to support the weight of everything above.

We also know that it is significantly hotter and quite likely to have some sorting of matter based upon density (and we know the composition is not the same from magma which has leaked to Earth's surface).

We also have strong evidence indicating that there is a molten core from Earth's magnetic field. Earth's magnetic field changes over time including flipping. This matches spinning balls of molten metal which undergo convection. A solid core wouldn't explain it.

But we don't have much in the way of possibilities of investigating what the structure of Earth is.

If you have any suggestions I am sure the geologists would welcome it.

And the idea of what's "flat" in non-Euclidean space is actually pretty well-defined, just as it's well-defined to say that a geodesic on a curved surface is "straight".

It can actually be a lot more complicated than that.

There is really only one way to define a straight line. You have a point and you have a direction. The curvature of space just means you follow that and get a geodesic. You can start at any point along the line and go forwards and backwards to make the rest of it.

But there are many ways to define a plane. One simple idea is a point, normal, which works in 3D space. You take a point and then consider all lines passing through that point which are normal to the normal.

Depending upon the geometry of the space, you can move along the surface, take a new point and the vector normal to that and construct a surface and get a different surface.

You can also construct a plane by taking a line and a vector and translating the line along that vector. Depending upon the geometry of space and what vector you choose you can get different surfaces. You can also pick another line on the surface and another vector and use that and generate a different surface.

This can also be interpreted as the same surface looking flat from one perspective (e.g. one point and way of measuring flatness) but curved from another.

For a flat space, all the planes constructed are the same. For a non-flat space, they will not necessarily be the same.

This makes the concept of a flat surface not as easy to move to a non-flat space as the concept of a straight line.

The other issue is that unlike a line, we can measure the curvature of a surface (at least if it is of constant curvature) with gaussian curvature, which gives a result independent of the space the surface is embedded in.

i.e. a curved surface in flat space would show its curvature, a "flat" surface in curved space would show the curvature of the space.

But yes, the difference between this hypothetical Earth and a RE in flat space would be huge. This hypothetical Earth wouldn't have a horizon. If you were high enough you would be able to observe the ground repeating. If you looked up you would see multiple images of the sun (and other stars) from the light "circling" Earth.