The Funk and Hilbert geometries for spaces of constant curvature

Abstract

The goal of this paper is to introduce and to study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets of the hyperbolic space $\mathbb H ^n$ and of the sphere $S^n$. We highlight some striking similarities among the three cases (Euclidean, spherical and hyperbolic) which hold at least at a formal level. The proofs of the basic properties of the classical Funk metric on subsets of $\mathbb R ^n$ use similarity properties of Euclidean triangles which of course do not hold in the non-Euclidean cases. Transforming the side lengths of triangles using hyperbolic and circular functions and using some non-Euclidean trigonometric formulae, the Euclidean similarity techniques are transported into the non-Euclidean worlds. We start by giving three representations of the Funk metric in each of the non-Euclidean cases, which parallel known representations for the Euclidean case. The non-Euclidean Funk metrics are shown to be Finslerian, and the associated Finsler norms are described. We then study their geodesics. The Hilbert geometry of convex sets in the non-Euclidean constant curvature spaces $S^n$ and $\mathbb H ^n$ is then developed by using the properties of the Funk metric and by introducing a non-Euclidean cross ratio. In the case of Euclidean (respectively spherical, hyperbolic) geometry, the Euclidean (respectively spherical, hyperbolic) geodesics are Funk and Hilbert geodesics. This leads to a formulation and a discussion of Hilbert’s Problem IV in the non-Euclidean settings. Projection maps between the spaces $\mathbb R ^n, \mathbb H ^n$ and the upper hemisphere establish equivalences between the Hilbert geometries of convex sets in the three spaces of constant curvature, but such an equivalence does not hold for Funk geometries.

Appendix

1.1 Some results in projective geometry on $\mathbb H ^n$ and $S^n$

We recall the following well-known formulae for the purpose of highlighting the analogy (at the formal level) between the three geometries, and because they are used in this paper.

Proposition 5.1

(sine rule) Given a triangle $ABC$ with sides $a,b,c$ opposite to the angles $A,B,C$ respectively, we have, in the case where the triangle is Euclidean:

$$\begin{aligned} \frac{a}{\sin A} =\frac{b}{\sin B}= \frac{c}{\sin C}, \end{aligned}$$

in the case where the triangle is spherical:

$$\begin{aligned} \frac{\sin a}{\sin A} =\frac{\sin b}{\sin B}= \frac{\sin c}{\sin C}, \end{aligned}$$

and in the case where the triangle is hyperbolic:

$$\begin{aligned} \frac{\sinh a}{\sin A} =\frac{\sinh b}{\sin B}= \frac{\sinh c}{\sin C}. \end{aligned}$$

For the proof, and for other trigonometric formulae in hyperbolic trigonometry we refer the reader to [1] where the proofs are given in a model-free setting. In such a setting the proofs work as well in the spherical geometry case.

We now state a few results in hyperbolic geometry that lead to the hyperbolic cross ratio invariance. We omit the proofs; they based on the sine rule and they mimic the proofs of corresponding results in Euclidean geometry. Analogous results hold in spherical geometry, where the hyperbolic sine function is replaced by the sine function.

Proposition 5.2

Let $ABC$ be a hyperbolic triangle. We join $A$ by a geodesic to a point $D$ on the line $BC$. Then we have

$$\begin{aligned} \frac{\sinh DC}{\sinh BD}=\frac{\sin \widehat{DAC}}{\sin \widehat{BAD}}\cdot \frac{\sin B}{\sin C}. \end{aligned}$$

There proof can be done in applying several times the sine rule.

Proposition 5.3

Consider four ordered distinct geodesic lines $l_1,l_2,l_3,l_4$ intersecting at a point $A$ and let $l$ be a geodesic line intersecting $l_1,l_2,l_3,l_4$ at points $A_1, A_2, A_3, A_4$ respectively. Then we have

$$\begin{aligned} \frac{\sinh A_2 A_4}{\sinh A_3A_4}\cdot \frac{\sinh A_3A_1}{\sinh A_2A_1}= \frac{\sin \widehat{A_2AA_4}}{\sin \widehat{A_3AA_4}}\cdot \frac{\sin \widehat{A_3AA_1}}{\sin \widehat{A_2AA_1}}. \end{aligned}$$

Proof

We refer to Fig. 2. Applying Proposition 5.2, we have

$$\begin{aligned} \frac{\sinh A_2A_4}{\sinh A_3A_4}= \frac{\sin \widehat{A_2AA_4}}{\sin \widehat{A_3AA_4}}\cdot \frac{\sin A_3}{\sin A_2} \end{aligned}$$

and

$$\begin{aligned} \frac{\sinh A_3A_1}{\sinh A_2A_1}= \frac{\sin \widehat{A_3AA_1}}{\sin \widehat{A_2AA_1}}\cdot \frac{\sin A_2}{\sin A_3}. \end{aligned}$$

From these two equations, the statement follows. $\square$

Fig. 2

Cross ratio invariance

As a consequence, we have the following

Corollary 5.4

(Cross ratio invariance in hyperbolic geometry) Consider four ordered distinct geodesics $l_1, l_2, l_3, l_4$ in the hyperbolic plane that are concurrent at a point $A$ and let $l$ be a geodesic that intersects these four lines at points $A_1, A_2, A_3, A_4$ respectively. Then the cross ratio of the ordered quadruple $A_1, A_2, A_3, A_4$ does not depend on the choice of the line $l^{\prime }$.

We now state the following hyperbolic analogue of a classical theorem due to Menelaus in the spherical case. This theorem and its analogue in spherical geometry can be used to give an alternative proof of the triangle inequality for the hyperbolic and spherical Funk metrics. The reader can follow step by step the proof of the triangle inequality given in [23] replacing lengths of segments by the hyperbolic sine of this length (in the hyperbolic case) of by the sine of this length (in the spherical case).

Theorem 5.5

(Menelaus’ Theorem in the hyperbolic plane) Let $ABC$ be a triangle in the hyperbolic plane and let $A^{\prime },B^{\prime },C^{\prime }$ be three points on the lines containing the sides $BC, AC, AB$ (see Fig. 3). Then, the points $A^{\prime },B^{\prime },C^{\prime }$ are aligned if and only if we have the relation

$$\begin{aligned} \frac{\sinh AC^{\prime }}{\sinh AB^{\prime }} \cdot \frac{\sinh BA^{\prime }}{\sinh BC^{\prime }}\cdot \frac{\sinh CB^{\prime }}{\sinh CA^{\prime }}= 1 \end{aligned}$$

Fig. 3

Menelaus’ theorem

The proof is an application of the hyperbolic sine rule.