No, space is not curved

Space is flat.  What does this mean?  It means that the three familiar dimensions of space (x, y, z) obey the Pythagorean principle for Euclidean space.

0 ≤ (x2 + y2 + z2)  ≤ c2 

But what does this mean?  x, y, and z are not deformations, but velocities, so they have an intrinsic time element which we usually ignore.  No, it's not proper (subjective) time.  This is the home of true, objective time.  Yes, it exists, because it must.  After all, light propagates and changes over time, without experiencing any time itself.  But it is routinely ignored, for reasons which completely escape me.  (Note that the axes of a spacetime diagram are true distance and true time.  But we, for some reason, must pretend that these don't exist.  Possibly because we, being made of matter, can never actually experience them?)

It would be easier and more intuitive to replace the awkward x, y, z with a simple v for velocity, ignoring the actual direction, which is immaterial for this narrow purpose.  

0 ≤ v2 ≤ c2 

Of course, nothing with mass can travel at the speed of light.  By similar logic and a careful examination of the meaning of quanta, it cannot remain perfectly stationary, either.  To do so, the particle must give up that last bit or momentum.  But there is no way to drop that last bit, because it is less than the smallest quanta of energy transfer.  So we may simplify the expression slightly for normal matter.  (Notice that a photon either travels at the speed of light, or stops existing, but never anything in between.)

0 < v2 < c2

Spacetime, on the other hand, is not flat, but hyperbolic.  This is because proper (perceived, subjective) time is imaginary, and when you square it, out pops a minus one.

0 < (v2 - t2) < c2 

Oddly enough, you can negate the central term, and it still holds true.

0 < (t2 - v2) < c2