In 2 dimensions, Euclidean geometry is the geometry of a flat piece of paper. But there are also of course curved 2-dimensional surfaces. The simplest of these is the sphere, which has constant positive curvature and is a model of (double) elliptic geometry. Another important example is the hyperboloid, which also has constant curvature, and is a model of hyperbolic geometry. Hyperbolic and elliptic geometry form the 2 main categories of non-Euclidean geometries.

In fact, any 2-dimensional surface in Euclidean 3-space provides a possible geometry, most of which are curved. However, it is important to realize that distances are always positive in all such geometries. One measures the distance between 2 points on such a surface by stretching a string between them along the surface. This does not measure the (3-dimensional) Euclidean distance between the points, but instead corresponds to integrating the arclength along the shortest path between them.

In hyperbola geometry, we instead made a fundamental change to the distance function, allowing it to become negative or zero. If there is precisely one (basis) direction in which distances turn out to be negative, such geometries are said to have Lorentzian signature, as opposed to the Euclidean signature of ordinary surfaces. As implied by the way we have drawn it, hyperbola geometry turns out to be flat in a well-defined sense, which immediately raises the question of whether there are curved geometries with Lorentzian signature.

The mathematical study of curved surfaces forms a central part of differential geometry, and the further restriction to surfaces on which distances are positive is known as Riemannian geometry. The much harder case of Lorentzian signature is known, not surprisingly, as Lorentzian geometry, and the important special case where the curvature vanishes is Minkowskian geometry. This classification of geometries by signature and curvature is summarized in Table 13.5.

What does this have to do with physics? We have seen that hyperbola geometry, more correctly called Minkowski space, is the geometry of special relativity. Lorentzian geometry turns out to be the geometry of general relativity. In short, according to Einstein, gravity is curvature!

Table 13.5: Classification of geometries.