Curved space or curved spacetime?

As I understand it, you can have time + flat space = curved spacetime.

1. So, when one is trying to emphasise that there is a curvature to the space, is it more technically correct to say curved space in favour of curved spacetime.
2. Does anyone do this?

It can be curved though, depending on how you join the two to form a spacetime. For example, flat space can be joined with a time coordinate to form a hyperbolic space with metric $$ds^2 = \frac{-dt^2 + dx^2 + dy^2 +dz^2}{t^2},$$ such that for every $t$ fixed, we get a flat spacial slice, but the total space is most definitely curved.