You evaded my question, Son Goku.
Is it not more plausible that I simply misunderstood it or that you misunderstood me? I must say this is a very "rude" claim to make.
But coupling to space and matter is what (abe: those properties) allegedly allow space to do, not what those properties are. An analogy would be, if you join a team, (abe: being a member of a team) is not a property of you as a human, it is something; an activity that you, (abe: the human,) has decided to join up to.
Note: I haven't forgotten your question but I need you to answer mine forthrightly first. Fair enough?
So we're back to square one. Son Goku, what properties of space allow for it to be curved/warped by energy and matter? I assume a forthright answer would be we really don't know. Do you agree with Onifre that this is correct?
Okay, the real answer. Although it will not be much use.
The diffeomorphism invariance of spacetime is the property that allows it to curve. This provides a conserved current, the Stress-Energy tensor which matter can then couple to. This allows matter to directly influence the Ricci curvature of spacetime.
If this does not satisfy you then it's the fact that spacetime is pseudo-Riemannian.
Or rather the question has a much simpler answer given already. Which is that it can curve because it can curve. There is no need for me to evade. Spacetime's curvature has been measured, that is a simple fact. There is no need to understand curvature in terms of other underlying properties, it is a property itself.
Edited by Son Goku, : Addition.