Distinction between spacetime and vacuum In GR, is spacetime just a mathematical abstraction, and in reality, it's the vacuum - whatever that is!!- is what curves, bends, and warps? In other words, is the distinction between spacetime and vacuum of physical or mathematical nature?
 A: Vacuum is a particular state in a vector space, which is defined in terms of quantum mechanics, while spacetime is a whole topological (or vector) space, defined in terms of general (or special) relativity. 
This means that spacetime is a classical phenomenon while vacuum is a quantum phenomenon. However, there is also a vacuum state in classical physics, but it is a thermodynamical state where there isn't any content at all, that's why in classical physics it is sometimes called free space.
In quantum mechanics, the vacuum state is represented by a vector, $| \Omega \rangle$ or $| 0 \rangle$, in a Hilbert space, and it has a non-zero energy, but the avarage number of particles is zero even if it fluctuates by virtual particles popping into existence and vanishing all the time. In classical thermodynamics, the vacuum state is represented by a particular configuration where all the content that has energy or momentum is removed. Therefore, in general relativity, it corresponds to a vanishing energy-momentum tensor, however it does not necessarily mean the curvature of spacetime is flat, i.e., the cosmological constant can contribute to the spacetime curvature even it is in a vacuum state. 
Note that spacetime is a classical object to our most recent knowledge. If a model of a quantum gravity theory comes into play, then we would define a quantum spacetime as well. But for now, even in quantum mechanics, the spacetime is classical (and flat). 
More precisely, spacetime is a topological space endowed with a metric field and a connection, where other fields, either classical or quantum, can have values at each of its elements. When those fields have all their (expectation) values at all points in a region of spacetime set to zero, then it is called the vacuum.
A: It is really the coordinates (x,t) of things in spacetime that define its geometry. For example, a test object moving in space (vacuum) where there is gravity traces out a curved trajectory as time passes. 
This (x,t) trajectory of free fall in vacuum is a geodesic of a 4-dimensional manifold M. This manifold is then called 'spacetime', abstracted from the test object tracing it out. So, where there is gravity, empty space and time are not really 'bent and warped', but the trajectories of falling objects in it are.
