Riemannian Geometry – Lecture 20 Isotropy Continued

Dr. Emma Carberry

October 12, 2015

Hyperbolic Space: upper half space model

Example 20.1. The upper half plane model for hyperbolic space of radius r is given by

n 1 n n n U (r) = {(x , . . . , x ) ∈ R | x > 0} together with the Riemannian metric deﬁned in coordinates (x1, . . . , xn)

r2 gU (x1, . . . , xn) = δ ij (xn)2 ij where we are taking the standard basis vectors ∂i = ei = (0,..., 0, 1, 0,..., 0).

A computation very similar to that we performed earlier shows that this has constant sectional −1 curvature r2 .

Example 20.1 (Continued).

Hyperbolic Space: ball model

1 Example 20.2. The ball or disc model for hyperbolic space of radius r is given by

n 1 n n 1 2 n 2 2 B (r) = {(x , . . . , x ) ∈ R | (x ) + ··· + (x ) < r } together with the Riemannian metric deﬁned in coordinates (x1, . . . , xn) by

4r4δ gB(x1, . . . , xn) = ij . ij (r2 − |x|2)2

Example 20.2 (Continued).

Hyperbolic Space: hyperboloid model

Example 20.3. The hyperboloid model for hyperbolic space of radius r is given by

n−1,1 1 n n+1 n,1 S (r) = (x , . . . , x , x ) ∈ R | (xn+1)2 − (x1)2 − · · · (xn)2 = r2, xn+1 > 0 together with the Riemannian metric gH = ι∗ρ induced from the inclusion ι : Sn−1,1(r) → Rn,1. Here Rn,1 is the manifold Rn+1 together with the Minkowski metric

ρ(x, y) = x1y1 + ··· + xnyn − xn+1yn+1.

Note that ρ is not a Riemannian metric so the fact that gH is Riemannian (the hyperboloid is “spacelike”) needs to be checked.

2 x^(n+1)

x^2

x^1

Proposition 20.4. These three models of hyperbolic space are all isometric Riemannian mani- folds.

An isometry from the ball model to the upper half plane model is given by, with 1 ≤ i ≤ n − 1:

f : Bn(r) → Un(r) 2 i 2 1 2 n 2 (. . . , xi, . . . , xn) 7→ ..., 2r x ,..., r(r −(x ) −...−(x ) ) , (x1)2+···+(xn−1)2+(xn−r)2 (x1)2+···+(xn−1)2+(xn−r)2 which is a generalisation of the Cayley transform of complex analysis.

An isometry from the hyperboloid model to the ball model is given by hyperbolic stereographic projection.

We omit the check that these are indeed isometries.

3 We write (Hn(r), g) for any one of the model spaces deﬁned above and call it hyperbolic space of radius r.

It is also a homogeneous and isotropic Riemannian manifold which is most easily seen with the hyperboloid model.

Let O(n, 1) denote the group of invertible linear transformations Rn,1 → Rn,1 which preserve the Minkowski metric ρ. This is called the Lorentz group.

Then the Lorentz group preserves the hyperboloid of two sheets

1 n n+1 n,1 (x , . . . , x , x ) ∈ R | (xn+1)2 − (x1)2 − · · · (xn)2 = r2 and the subgroup of it preserving the upper sheet

n−1,1 1 n n+1 n,1 S (r) = (x , . . . , x , x ) ∈ R | (xn+1)2 − (x1)2 − · · · (xn)2 = r2, xn+1 > 0 is denoted by O+(n, 1).

n−1,1 Proposition 20.5. The positive Lorentz group O+(n, 1) acts transitively on S (r) and more- over acts transitively on the set of orthonormal bases on Sn−1,1(r). Hence hyperbolic space Hn(r) is an isotropic homogeneous Riemannian manifold.

The proof is similar to that for the sphere.

Corollary 20.6. Hyperbolic space Hn(r) has constant sectional curvature.