Just to avoid any confusion, it may be worth noting that a "normal" (three dimensional) sphere (or torus) is a 2-sphere (2-torus). I assume this because mathematicians are only concerned with the two-dimensional surface of such an object. Which is also related to previous discussions here about a circle being a one dimensional object.
Just a side note on negative curvature. I've seen it more as a mathematical tool to find the maximum and minimum of a function. If the function 'flatlines' at point A, that could be the maximum or minimum of the function. The easiest way to do that is to look at the curvature. For a function of multiple variables, positive curvature would correspond to said max/min. Negative curvature is called a saddle point.
For example, if y=x2 , we all know from experience that there is no maximum, but a minimum at the point (x,y)=(0,0).
Now try y=x3. Its derivative for x=0 is the same, 0, However, that is a type of saddle point. It is neither the maximum or minimum of the function. As you approach the origin from the left, the curvature is 'downwards'. As you approach the origin from the right, it curves 'upwards'.
An easy way to visualize it is to tell yourself, x2 keeps wanting to curve downwards. Since it always curves in the same direction, it is positive curvature. However, x3 changes from curving 'downwards' to curving 'upwards'. This flip in curvature, combined with a derivative property(italic so I have ammo when someone jumps on that even though I haven't omitted it) is negative curvature. In 3D, you have the basic saddle which curves say, upwards in the x direction, and downwards in the y direction. That's not important right now.
Things like sine waves obvious flip between 'upwards' and 'downwards' curvature, but the derivative must 0 to qualify as a saddle point. That derivative property is called being a 'stationary point'. If the function represents a force, there would be no force at that point, the object would remain stationary.
All signs point to a flat universe, so there's no point in working with a negative curvature spacetime for the universe.
But as I understand it you can't have a flat finite universe with matter in it without some negative curvature some where. Is there another manifold that allows for a finite unbounded flat universe without negative curvature?