We build two n dimensional uniformly curved spaces embedded within an n+1 dimensional vector space over the reals. The first, a sphere, is positively curved and the second, one sheet of a hyperboloid, is negatively curved. For each of these we describe two flat models:
We require a quadratic form q on our vector space which is especially chosen for the case at hand.
Coordinates for our vector space are x = <t, x_{1} ... x_{n}>.
For the following equations j ranges over 1 ... n.
For the sphere the quadratic form is q(x) = t^{2} + (Σ_{j}x_{j}^{2}). (Euclidean for n=2 or 3)
For the hyperboloid the quadratic form is q(x) = t^{2} − (Σ_{j}x_{j}^{2}). (Minkowskian for n=3)
In each case the embedded n dimensional curved space H is the set of vectors x such that q(x) = 1, but we restrict ourselves to t > 0 for the hyperbolic case.
Minkowski space was invented for relativity.
Here is a connection.
In each case the metric for the curved space H is derived from the quadratic form. For Euclid this is the normal pythagorean metric for both the vector space and the sphere. For the Minkowski space the pseudo metric is as introduced by Minkowski to describe relativity. In either case the distance between nearby points x & x' in the embedded space is √q(x − x') for the case of the sphere, and √−q(x − x') for the hyperbolic case. For both sorts of curved space the local metric in the space is positive definite. We still postpone curved Minkowski manifolds.
A metric must yield a positive real except that δ(x, x) = 0. Thus the ‘pseudo’ in ‘pseudo metric’. Quadratic forms may be positive or negative.
For any quadratic form there is a group of linear transformations that preserve that form.
Given a quadratic form q with signature (+, −, − ... −), such as we have chosen to define our hyperbolic space, there is a similar group a transformations which preserve that form. If η is a diagonal matrix with with the signature values down the diagonal then the group of matrices A such that AηA^{t} = η preserves our form.
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. There are six generators for this group for n=3. They are:
1 | 0 | 0 | 0 |
0 | cos θ | sin θ | 0 |
0 | −sin θ | cos θ | 0 |
0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 |
0 | cos θ | 0 | sin θ |
0 | 0 | 1 | 0 |
0 | −sin θ | 0 | cos θ |
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | cos θ | sin θ |
0 | 0 | −sin θ | cos θ |
cosh θ | sinh θ | 0 | 0 |
sinh θ | cosh θ | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
cosh θ | 0 | sinh θ | 0 |
0 | 1 | 0 | 0 |
sinh θ | 0 | cosh θ | 0 |
0 | 0 | 0 | 1 |
cosh θ | 0 | 0 | sinh θ |
0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 |
sinh θ | 0 | 0 | cosh θ |
This program generates such matrices randomly.
(Set M in the program to our n+1.
Set sig = 0 for spherical case, and = 1 for hyperbolic case.)
Here is another program that generates such matrices using Clifford Algebras.
The spherical conformal case is the Riemann sphere where the plane is normally taken to be the plane of complex numbers. The hyperbolic space is infinite both in its embedding and intrinsically, but its two models are finite. The spherical space is finite but its two models are infinite.
Sphere | Hyperboloid | Projection center | |
Geodesic | Projective | Klein | <0, 0, 0, ...> |
Conformal | Riemann | Poincaré | <−1, 0, 0, ...> |
pictures |
Theorems from Euclidean geometry can be applied locally to curved spaces via conformal models. Theorems from projective geometry can be applied via the geodesic maps.
Here is a different flat model of curved Minkowski space with different charms and cosmological pretensions.
1 | 0 | 0 | 0 |
xd | 1 | 0 | 0 |
yd | 0 | 1 | 0 |
zd | 0 | 0 | 1 |