An isometry is a map from a metric space to itself that preserves distances. It turns out that the set of isometries for a particular uniformly curved space is simpler than for the familiar flat Euclidean space, at least by the tools we pursue here.
Spaces with uniform curvature have either positive or negative curvature. The math is almost the same but there are subtle differences. We also describe in both cases, two flat maps of curved spaces. Conformal maps preserve angles, and geodesic maps preserve geodesics. See this marvelous app for a tour of such spaces.
You might want to follow a slightly more formal and uniform presentation.
The negatively curved hyperbolic space is less familiar but only slightly more difficult. Instead of embedding in an n+1 dimensional Euclidean space (with its Pythagorean distance) we must choose an n+1 dimensional Minkowski space where the (distance between two points)2 is δt2−δx2−δy2−δz2 for the n=3 case. We choose a point in this space and call it the origin. The distance2 of a point from the origin is a quadratic form whose signature is (+ − − −) for (1, 3) or generally (1, n). Minkowski invented this space for relativity; here is a connection.
Consider the set of points H where t>0 and whose pseudo distance from the origin is 1. H is an n dimensional subspace (but not linear subspace) of the Minkowski space. The metric on H induced by the pseudo metric is a true metric and the intrinsic curvature of H by that metric is uniformly −1. H is much like the sphere in the previous example and there is a set of matrices that act on the Minkowski space and leave points of H in H. These are exactly the isometries of H, no more-no less.
What are these matrices that leave H in H? The condition for A to be such a matrix is that ||Ax|| = ||x|| for all x, where ||x|| is the length of vector x by the pseudo metric. These matrices form a Lie group under multiplication. They have been called “signature orthogonal” but perhaps now more commonly “Indefinite orthogonal groups”. There are three generators for this group for n=2. They are:
|0||cos θ||sin θ|
|0||−sin θ||cos θ|
|cosh θ||0||sinh θ|
|sinh θ||0||cosh θ|
|cosh θ||sinh θ||0|
|sinh θ||cosh θ||0|
If we choose P at the center of the sphere instead of the surface we have a different interesting map from half of the sphere to the plane. This relation maps lines in the plane to geodesics (great circles) on the sphere.
If we project from P = <0, 0, ...> to H we pass thru the plane T where t=1. The unit disc (ball) in T maps to all of H. Geodesics in H map to lines in T and conversely. This is the Klein map of the hyperbolic space.
We tile curved spaces here.
Tomorrow, the more difficult flat space.