Hilbert geometries with Riemannian points

If a Hilbert geometry of twice differentiable boundary has two quadratic infinitesimal spheres, then the Hilbert geometry is a Cayley–Klein model of the hyperbolic geometry.


Introduction
For the proof, we need the assumption that the boundary is twice differentiable at the points, where the line joining the two Riemannian points intersects the boundary. Theorem 5.2 shows that this assumption is also necessary. Theorem 4.4 is also formulated in the language of geometric tomography [7] by Theorem 5.3: the twice differentiable boundary of a strictly convex bounded domain in the plane is an ellipse if and only if its (−1)-chord functions are quadratic at two inner points.

Notations and preliminaries
Points of R n are denoted by capital letters A, B, . . ., vectors are − → AB or a, b, . . ., but we use these latter notations also for points if the origin is fixed. We denote the interior of the convex hull of a point set P by P.
For C ∈ AB, the affine ratio If a Euclidean metric d e is given, then the length of a segment AB, or of a vector − → AB = x is denoted by |AB| = |x| = d e (A, B).
We use the usual big-O and little-o notation. To indicate derivatives of a function or a map, we use prime, dot or D appropriately.
If the domain M of the Hilbert geometry (M, d) is in R n , then we identify the tangent spaces T P M with R n by the map ı P : v → P + v. This way, the Finsler function F M : M × R n → R associated with the Hilbert metric d can be given at a point P ∈ M by [4, (50.4)]. 1 Equation (2.1) implies that ı P maps the indicatrix of norm F M (P, ·) into the strictly convex set B M P ⊂ R n , the infinitesimal ball, with boundary the infinitesimal sphere. Observe here that if is a projective transformation on the projective completion P n of R n , then its derivative˙ is an affine transform from each tangent space From now on, we work only in the plane unless explicitly said otherwise.
So, infinitesimal spheres are called infinitesimal circles and denoted by C M P . If a Euclidean metric is provided, then we frequently use the notation u ϕ = (cos ϕ, sin ϕ). Further, if a bounded open domain D ⊂ R 2 is starlike with respect to a point P ∈ D, then we usually polar parameterize the boundary ∂D with a function r : [−π, π) → R 2 defined by r(ϕ) = r (ϕ)u ϕ ∈ ∂D, where r > 0 is the radial function of D with respect to the base point P. For any ellipse E with center P there exists unique ω ∈ (−π/2, π/2] and a ≥ b > 0 such that 1 is the polar equation with respect to origin P. We also use the notation d := {λd : λ ∈ R} for the line through the origin with nonvanishing directional vector d, and ξ = u ξ as a short hand in the plane.
The following result is a rephrase of [5,Stable Manifold Theorem,p. 114]. See also [6,Theorem 4.1]! Theorem 2.1 Let N 0 ⊂ R 2 be a neighborhood of the origin 0, and let the mapping If there are linearly independent vectors u and v such that (w) = w for every w ∈ u ∩ N 0 , and D (0,0) v = kv for some k ∈ (0, 1), then in some neighborhood N ⊆ N 0 of 0 the set {w ∈ N : (r ) (w) → 0 as r → ∞} is the graph of a C l function from v ∩N to u ∩N .
Notice that (r ) refers to the r -th iterate, rather than, e.g., the r -th derivative. Finally, we need the following easy consequence of [4, (28.

Utilities
Although it is known that the hyperbolic geometry is a Riemannian manifold, so its infinitesimal spheres are quadratic, the following result gives some more details.
From now on, we always use the following general configuration: P is a point of a 2dimensional Hilbert geometry (M, d); is a straight line through P; I and J are the points where intersects ∂M; a coordinate system is chosen 2 such that I = (−1, 0), J = (1, 0), and P = ( p, 0), where −1 < p < 1; X and Y are the points where P + ξ intersects ∂M.  Observe that for X ∈ ∂M we have 2F M (P, X − P) − 1 = 1/λ − X −P > 0 by (2.1), so, as a continuous function takes its minimal value, there is a suitably small ε > 0 such that the map is well defined on the Minkowski sum M ε := ∂M + εB 2 , where B 2 is the unit ball at (0, 0). Choose the Euclidean metric d e such that {(1, 0), (0, 1)} is an orthonormal basis, and polar parameterize C M P with respect to P by r : .
Thus r is twice differentiable if ∂M is twice differentiable, and

4)
and . To prove (3.5), we are estimating −u for the second order of x and y. We start with (3.2) and use (3.3) as shows. Next we estimate |X P| by the binomial series so that Substitution of this into the previous formula and some rearrangements result in To estimate this, we need to consider r (ξ ) − r (0) and 1/(2|X P| − r (ξ )). We use the binomial series and (3.6) to get This, as sin ξ = −y/|X P|, leads to Substitution of this into the Taylor expansion of r gives Again the binomial series, and then (3.3), (3.6), and (3.9) result in (3.10) Putting estimates (3.9), (3.10), and (3.8) into (3.7) and confining ourselves to summands of degree less than three, we obtain where the summands that are estimated by O(x 3 ) + O(x 2 y) + o(y 2 ) was left out. Collecting the terms by their powers gives This implies (3.5) after reordering the summands.

Hilbert geometries with two Riemannian points
In what follows, we always assume that P and Notice that the infinitesimal circle C M P is now an ellipse, so it is of form (2.3) in any Euclidean metric. Observe that differentiation of (2.3) yields Further, using we obtain  Figure 2 shows what we have.

(4.2)
Assume from now on that X ∈ ∂M, hence also Y = P (X ) ∈ ∂M.
Since t I and t J are perpendicular to line Q P, basic differential geometry gives that the respective curvatures of ∂M at I and J are . (4.4) Repeating the same procedure for the circle C M Q gives κ J = −κ I + 2 1−q 2 . This and (4.4) imply hence Lemma 3.1 proves the statement with the ellipse x 2 + y 2 1−q 2 = 1.

Lemma 4.2 If ∂M is twice differentiable at I and J , then E coincides ∂M in a neighborhood of I , J , respectively.
Proof According to the last formula in the proof of Lemma 4.1, the infinitesimal circles , respectively. Then (3.1) gives hence P is a real analytic map on M ε . It follows in the same way that Q is a real analytic map on M ε . We conclude that := Q • P is also a real analytic map on M ε .

Lemma 4.3 If two Hilbert geometries have two common Riemannian points Q and P, and their borders coincide in some neighborhood of line P Q, then the two Hilbert geometries coincide.
Proof Let (L, d L ) and (M, d M ) be Hilbert geometries with common Riemannian points Q and P. Assume that there is a neighborhood N of line P Q that intersects the border of our Hilbert geometries in two common arcs I 0 and J 0 . Let line P Q intersect I 0 and J 0 in points I and J , respectively. We can assume without loss of generality that the points are ordered as I ≺ Q ≺ P ≺ J . So, we can use the notations already introduced in this paper.
Observe that C L Q ≡ C M Q and C L P ≡ C M P , because the common arcs of ∂L and ∂M determine small common arcs of the quadratic infinitesimal circles near line Q P. Thus both P and Q map any common arc of ∂L and ∂M to a common arc of ∂L and ∂M. We generate common arcs by defining J k+1 := Q (I k ) and I k+1 := P (J k ) for every k = 0, 1, . . .. Let α k (k = 0, 1, . . .) be the angle I k subtends at Q, and let β k (k = 0, 1, . . .) be the angle J k subtends at P.
To show that it is contradictory, assume that every α k and β k (k = 0, 1, . . .) is less than π. Then we clearly have β 0 < α 1 < β 2 < α 3 < · · · < β 2k < α 2k+1 < β 2k+2 < · · · < π. So I = lim k→∞ I 2k+1 subtends angle α = lim k→∞ α 2k+1 ≤ π, and J = lim k→∞ J 2k subtends angle β = lim k→∞ β 2k ≤ π. From the sequence of inequalities α = β follows, hence Q (I) = J and P (J ) = I. Then the assumption implies that α = β < π, which contradicts Q = P. So one of α k or β k (k = 0, 1, . . .) is at least π, say α k ≥ π. Then I k ∪ Q (I k ) covers ∂L and ∂M, and the lemma is proved. Falconer's [5,Theorem 4] gives that for any two fixed points P, Q several distinct strictly convex bounded open domains M exist in the plane such that P, Q ∈ M, the (−1)-chord functions at P and Q are equal to 1, the boundary ∂M is differentiable at I , J ∈ P Q ∩ ∂M and twice differentiable everywhere else, and ∂M is not an ellipse. Observe that in such an M there can not exist a third inner point with quadratic (−1)-chord function, because then ∂M has to be an ellipse by Theorem 5.1. Reformulating these to Hilbert geometries we obtain the following.

Theorem 5.2 Let d e be a Euclidean metric on the plane, and let C Q and C P be unit circles with centers Q and P, respectively.
Then there are several distinct non-hyperbolic Hilbert geometries (M, d) such that C Q and C P are the only quadratical infinitesimal circles in (M, d). The boundary of such a Hilbert geometry is twice differentiable except where it intersects line Q P.
How the Hilbert geometries given in this theorem relate to the hyperbolic geometry remains an interesting question.
Theorem 4.4 also raises the problem to determine those pair of ellipses that are infinitesimal circles of a Hilbert geometry. This can be done by following the proof of Lemma 4.1; the details remain to the interested reader for now.
One can specialize [7, Theorem 6.2.14, p. 247] to the following: This gives the following result which is more general, but weaker for the quadratical case than the combo of the lemmas in the previous section. Notice that this theorem states only a coincidence and therefore implies a weaker version of Theorem 4.4 only together with Lemma 4.1.
It is proved in [8, Theorem 2] that perpendicularity in a Hilbert geometry is reversible for two lines if the perpendicularity of these two lines is also reversible with respect to the local Minkowski geometry at the intersection of the lines 3