For this development we stress the similarity between constructions.

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, x1 ... xn>. For the following equations j ranges over 1 ... n.
For the sphere the quadratic form is q(x) = t2 + (Σjxj2). (Euclidean for n=2 or 3)
For the hyperboloid the quadratic form is q(x) = t2 − (Σjxj2). (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 positive definite form (q(x) > 0 for x≠0) and a basis which is orthonormal according to that form, the orthogonal matrices from O(n+1) define just the transformations that preserve q. Our chosen coordinate system is built on such a basis.

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 AηAt = 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

Each of these when operating on vector <t, x, y, z>, preserves t2 − x2 − y2 − z2. Thus vectors in H stay in H. These compose the subgroup of the Lorentz group that leaves the origin fixed. 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. Set sig = 0 for spherical case, and = 1 for hyperbolic case.)
Here is another program that generates such matrices using Clifford Algebras.

### The Flat Models

We project our embedded curved spaces onto the flat plane (n-space) defined by t = 1. For the geodesic models we project from <0, 0, 0, ...>; for conformal models we project from <−1, 0, 0, ...>. The geodesic hyperbolic case is the Klein model and the conformal hyperbolic case is the Poincaré model.

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.

The quadratic form q(x) = t2 is degenerate; H is the set where t = 1. The projected map is the identity. The linear transformations that leave t fixed have a top row = <1, 0, 0, ...> but not all of these are isometries on H. The first three generators given above for n=3 plus the following generate all of the transformations that are isometries on H:
 1 0 0 0 xd 1 0 0 yd 0 1 0 zd 0 0 1
where xd, yd, zd are a translation in the H space.