Not an expert, but: the curvature of spacetime is modeled as a tensor field (the metric tensor). That field can have (classical) waves in it, which is what LIGO detects (I believe). Then you can certain hypothetically quantize that field, in which case it definitely has to be a spin-2 particle and it seems likely that there will be a way to do it since all the rest were.
The "geometry" comes from the fact that the way we measure distances (or, well, experience time) uses the metric tensor field to do it. But it is still ultimately just a value attached to every point like any other field.
The "geometry" comes from the fact that the way we measure distances (or, well, experience time) uses the metric tensor field to do it. But it is still ultimately just a value attached to every point like any other field.