Isometries in Curved Space

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 Sphere

We start with the n-sphere which is positively curved. You won’t miss much if you think 2-sphere which is the surface of a familiar 3D ball. If we consider the n-sphere embedded in the flat n+1 dimensional vector space, with the sphere’s center at the origin, then the orthonormal transformations by orthogonal matrices from O(n+1) of the vector space provide the isometries on the sphere. (N.B. “Orthogonal” applied to a transformation is meaningful only in the context of a quadratic form, such as a metric.) An orthogonal matrix, operating on a vector leaves its length unchanged; thus it maps the sphere onto the sphere. Indeed any isometry on the sphere corresponds to some matrix in O(n+1).

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:

1	0	0
0	cos θ	sin θ
0	−sin θ	cos θ

cosh θ	0	sinh θ
0	1	0
sinh θ	0	cosh θ

cosh θ	sinh θ	0
sinh θ	cosh θ	0
0	0	1

Each of these when operating on vector <t, x, y>, preserves t²−x²−y². Thus vectors in H stay in H. These leave the origin fixed and compose a subgroup of the Lorentz group. In general there are n matrices with cosh and n*(n-1)/2 with cos. This program generates such matrices randomly. (Set M in the program to our n+1.)

Flat models of curved space

Flat Maps of Spheres

Suppose that we choose a fixed point P on our sphere and an n dimensional plane in the embedding space that is tangent to the sphere at the point opposite P. A line thru P to a point Z on the plane pierces the sphere at some other point Q as well as P. This relates Z to Q. This relation maps the sphere to the plane conformally. This is the Riemann sphere image of the plane. Sometimes the plane is taken thru the center of the sphere instead of tangent to it. Lines on the plane transform to circles on the sphere that pass thru P. Other circles on the sphere go to circles on the plane. Similarly for spheres of lesser dimension on our original sphere.

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.

Flat Maps of Hyperbolic Space

If we project from P = <−1, 0, ...> to H we pass thru the plane T where t=1.

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.

Refer somewhere to Hyperbolic space and Anti-de Sitter space.