Universal Constants in 3-Dimensional Hyperbolic Manifolds.

The topic of this Bachelor’s thesis are 3-dimensional hyperbolic manifolds. These are all 3-dimensional manifolds, with an hyperbolic metric, meaning a metric such that the space is negatively curved at every point and in very direction. These spaces are in some sense the strangest of the standard three geometries: Euclidean, spherical and hyper- bolic.

Usually it is hard to explain to a person, what the defining factor for a space being hyperbolic is. Euclidean geometry is easy to describe, because we are confronted with it regularly, as the space we live in, is at least locally Euclidean. (The global geometry of the universe is not yet known.) Also there are other well known Euclidean shapes like a flat torus, which everyone familiar with a bit of differential geometry can explain. With spherical geometry it already becomes a bit harder, but at least in dimensions one and two there are vivid examples, the circles and spheres. Here intuition can still guide us pretty far, since we are already somewhat used to the geometry of spheres because of the spherical shape of earth’s surface.

In contrast to those, there seems to be no space lending itself to hyperbolic geometry quiet in the same way as for the other two. A problem with visualizing hyperbolic spaces is their immense growth. Conversely to a circle on a sphere, where the the area of the circle grows less with the radius than in Euclidean space, the area a hyperbolic circle contains, grows exponentially with its radius. So there is not enough space to show a hyperbolic plane in Euclidean 3-space without it ”crinkling up”. Interestingly though, such crinkled planes can be seen in nature more than a few times for example with some jellyfishes, sponges and lettuces. So, also hyperbolic shapes appear in nature in some ways.