In 2D there is only scalar curvature, i.e. you don't need the whole Riemann tensor, just one number (at a given point).
In 3D you need the Ricci tensor, but still not the full curvature tensor. This is still much simpler, for instance IIRC this is why you cannot have gravitational waves in 3 (meaning 2+1) dimensional spacetime. In 4D you get the full complication.