If all matter were removed from space-time, what would be remaining? I would like to gain an understanding of the nature of the characteristics of space, as described in Einstein's theories. I appreciate that an answer using the language of mathematics is probably most applicable, but would not aid my comprehension in view of my deficit in this area. It would, therefore, be appreciated if responders could take this into account.  
For example, when, spacetime is described as being bent/warped/curved by a gravitational field created by a massive object, what is undergoing this deformation?  Is it only a mathematical construction that is being warped or is there something tangible that is undergoing this process of curvature?
Professor Brian Greene, in a YouTube presentation, described 'empty space'i.e. what was left after removing all matter (the dark matter was not discussed) as being cable of undergoing 'rippling', 'bending'' 'twisting'.  So my curiosity was naturally fuelled by his description.
Grateful for clarification, if this is possible.
 A: 
when, space-time is described as being bent/warped/curved by a gravitational field created by a massive object, what is undergoing this deformation? Is it only a mathematical construction that is being warped or is there something tangible that is undergoing this process of curvature?

All the phenomenons we have observed till date proves that it is sensible to state that General Relativity is not only a mathematical construct but there is indeed something tangible undergoing this process of curvature, but it totally depends on what you mean by tangible. You cannot ofcourse notice these effects in everyday life but on a very minute scale you can. Like the geometry you have probably studied in your school which is the Euclidean Geometry cannot hold true if you apply its rules to calculate geometrical measurements in your room, angles of a triangle would not add up to 180 degrees, the Pythagorean Theorem would not be true, as odd as it sounds but it is all due to the effect of warping or bending of space-time which changes the geometry of your room, because obviously, SPACE and time is being warped in the process due to the presence of mass of Earth and other objects. Remember not space alone is being warped, so the presence of mass warps TIME too, and so systems in space can evolve through time and change. Well, we cannot describe this warping more efficiently in words than it can be described mathematically, but I swear I would not include any calculations in it, just simple mathematical terms. So basically, there is a set of equations known as Einstein Field Equations that describe this warping of space-time and the change of Euclidean geometry mathematically. 

If all matter were removed from space-time what would remain?

Well if we remove all the mass as well as the energy from the space-time it would become flat because of no mass or energy present. The right hand side of Einstein Field Equations which has the Stress-Energy Tensor - that represent the amount of mass and energy at a give point in space-time would come out to be zero and so the total right hand side would be zero and so to equate that the left hand side would come out to be zero too. Therefore, there will be no warping of space-time, Euclidean Geometry and the geometry you have learned in your school will be correct. Any systems present would not evolve through time. I have provided a Relativistic Explanation since it is not possible to unify both Quantum Mechanical and Relativistic Explanations of gravity at this time.
A: Relativistically, spacetime itself has features: A general shape (topology) and a notion of (generalized pseudo-Euclidean) distance and angles (geometry).
This geometry can evolve in interesting ways even in the absence of matter, eg ripples in spacetime itself (gravitational waves).

Quantum mechanically, once we get rid of all field excitations (including particles), we're left with the vacuum.
Arguable, modern physics has resurrected a kind of aether, serving as medium for real particles similarly to how a crystal lattice acts as medium for phonons. In contrast to a crystal, the vacuum however is Lorentz-invariant.
Now, particles of a given field can interact with other fields even when they are in their ground state (vacuum polarization). Furthermore, fields may have a non-zero vacuum expectation value (cf the Higgs mechanism) as long as the vacuum state remains Lorentz invariant (thus the Higgs field is uniform across space). The lowest-energy zero-particle ground state is also not necessarily an eigenstate of the (appropriately smeared) field operator, ie measurements of field strengths may yield nonzero values even without any (real) particles being present (vacuum fluctuations).
A: I think the answer depends on what a singularity is (or if it exists). Kip Thorne has described a black hole as a soliton made of space-time. Yes, it has mass (and spin and charge), but does it have any matter? Perhaps they would be left behind in your hypothetical removal of all matter.
