If by spacetime you mean our Universe, then it is increasingly considered to be flat (http://www.universetoday.com/37029/flat-universe/) but that does not mean that it will collapse in on itself... quite the opposite... the universe will keep expanding, and the fate will likely be heat death / big-freeze where everything has the same temperature and no useful work can be carried out. Another scenario could be that the acceleration of the expansion becomes too much for EM and gravity, and everything breaks down to a thin soup of the most basic of building-blocks etc.so no1 thinks that spacetime will become flat and collapse in on itself because all matter has become stationary in relation to all other matter even though they are moving relative to space.
ah well i guess i got some deep thoughts and muttering to do.
A flat universe will not collapse in on itself.