The electromagnetic field is a function F(x), where F is the field tensor (which groups together the electric field E and the magnetic field B) and x is the position in space. Maxwell's equations tell you how F (or E and B) evolve in time, and give you wave-like solutions.

In a continuous medium like air, water, or a solid, you could describe a "displacement field" d(x) that tells you how far the piece of material that used to be at position x is displaced from its equilibrium position. This is governed by the equations of physical acoustics, which also have wave-like solutions: sound.

The Riemmann curvature tensor R is a function R(x) of spacetime coordinates x, which tells you the curvature of spacetime at location x. Einstein's equations tell you how R evolves, and it also turns out to have wave-like solutions: gravitational waves.

In radio, it's F (or E and B) that's waving. In gravitational waves, it's R.

What's confusing is that R also tells you "the shape of space," and you need to use it to figure out "where" the coordinate x corresponds to.