In general relativity, by contrast, the situation is very different. Einstein's equations tell us that matter curves space. Geometry is no longer immune to change. It reacts to matter. It is dynamical.
Geometry is a physical entity, very much like matter.
Now, the physics of this century has shown us that matter has constituents and the 3-dimensional objects we perceive as solids are in fact made of atoms. The continuum description of matter is an approximation which succeeds brilliantly in the macroscopic regime but fails hopelessly at the atomic scale. We know that a quantum theory of geometry should contain three fundamental constants of Nature, c, G, hbar, the speed of light, Newton's gravitational constant and Planck's constant. It is now called the Planck length. In this regime, the continuum picture works well.