What are positive and negative curvature when working in non-Euclidean geometry?

Question

Can you give me the sense I can find the difference between positive and negative curvature in non-Euclidean geometry better? I am asked today by my friend and I want to give her a good practical explanation. Thank you very much!

Answer 1

In spaces of positive curvature, triangles "bulge"; the sum of their angle measures exceeds 180 degrees. For visualization: Think of triangle on a sphere, with a vertex at the North Pole, and two on the Equator; there are right angles at the equator, and some-other angle at the Pole, so the angle sum is more than $180^\circ$.

In spaces of negative curvature, triangles "pucker"; the sum of their angle measures falls short of 180 degrees. For visualization: Take a look at the pseudo-sphere.

Answer 2

An example of a space of positive curvature is a sphere. An example of a space of curvature $0$ is a cylinder. To see the difference, try forming a cylinder from a rectangular piece of paper. There is no problem doing that, but try now to create a sphere out of a piece of paper. That won't work without distorting the paper, because the curvature changes. In more detail, if you cut off a small circular piece of the sphere and try to flatten it against a flat piece of paper it won't quite work, and it would feel as if the piece cut off from the sphere is missing material. Cutting the piece of sphere radially will help it fit (but not quite) over the flat piece of paper, but where the cut was made an angle will now form. The larger the angle the more curved the sphere was, and the more material was 'missing' from it, causing the positive curvature. You can also think of positive curvature as the result of gravitational pull, pulling material towards it, causing it to bend, but in such a way that from any point, no matter in which direction you look, you see the material bending towards the centre of gravity (unlike the cylinder, where there are still direction showing no bending at all).

Negative curvature is a bit trickier, but essentially it is the other way around to positive curvature. Instead of having too little material there is now too much. So if you were to cut off a small circular piece of the hyperbolic plane and tried to fit it over a flat piece of paper it won't quite work, but now it will feel as if you get overlaps of material. Making a radial cut in the piece of hyperbolic plane won't help, buy removing an angular segment will help, and you'll find that if you remove just the right amount (depending on the curvature of the hyperbolic plane you started with) you can pretty much (but not exactly) cover the flat circular piece of paper.

Of course, you can't really have any piece of hyperbolic plane to play with since it can't be embedded in $\mathbb R^3$, but there are some nice approximations. For example, a lattice leaf is a very rough approximation, it has only points exhibiting negative curvature. A more mathematical approximation is obtained as follows. Draw two concentric circles of radii about 20cm and 21cm (roughly). Cut the annulus and then chop it down radially into pieces of about 3-4cm long. Now, repeat with another 2-3 such annuli of the exact same radii as before. Now, glue the pieces back together but make sure not to just glue them radially (since that will just reconstruct the annuli) but instead, glue long side to short side of different pieces. If you do this while making sure to go in all directions you'll get a nice approximation of the hyperbolic plane. In fact, as the width of the annuli approach $0$ these approximations will actually approach the the hyperbolic plane.

Answer 3

I think if you want to explain this concept, you can use an orange. Indeed, a surface of an orange inside and outside of it could give us what the problem is about. But with geometry in view, when we are working with any spaces with positive curvature, similar to the surface of an orange, if you draw two greater circle, they intersect themselves so, there are no lines parallel to a given line through an outside point (Rejecting the 5-th postulate of Euclidean Geometry).

On the other hand, while in a space of negative curvature, like the surface of an hyperbolic paraboloid or inside the cover an orange , we can draw many lines parallel to a given line through an outside point (Again rejecting the 5-th postulate of Euclidean Geometry). What is so familiar to us happens when you are working on a blank paper. In fact, in this case, you are experiencing the zero curvature. In this case you can easily draw just a single line parallel to a given line through an outside point.

Now tell me what is the real curvature of our world? ;-)