Tits metric

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages) This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Tits metric" – news · newspapers · books · scholar · JSTOR (June 2018) (Learn how and when to remove this template message) This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (June 2018) (Learn how and when to remove this template message) This article needs attention from an expert in mathematics. The specific problem is: to review the article. WikiProject Mathematics may be able to help recruit an expert. (October 2019) (Learn how and when to remove this template message)

In

.

Ideal boundary of an Hadamard space

Let (X, d) be an Hadamard space. Two geodesic rays c₁, c₂ : [0, ∞] → X are called asymptotic if they stay within a certain distance when traveling, i.e.

${\displaystyle \sup _{t\geq 0}d(c_{1}(t),c_{2}(t))<\infty .}$

Equivalently, the Hausdorff distance between the two rays is finite.

The asymptotic property defines an equivalence relation on the set of geodesic rays, and the set of equivalence classes is called the ideal boundary ∂X of X. An equivalence class of geodesic rays is called a boundary point of X. For any equivalence class of rays and any point p in X, there is a unique ray in the class that issues from p.

Definition of the Tits metric

First we define an angle between boundary points with respect to a point p in X. For any two boundary points ${\displaystyle \xi _{1},\xi _{2}}$ in ∂X, take the two geodesic rays c₁, c₂ issuing from p corresponding to the two boundary points respectively. One can define an angle of the two rays at p called the Alexandrov angle. Intuitively, take the triangle with vertices p, c₁(t), c₂(t) for a small t, and construct a triangle in the flat plane with the same side lengths as this triangle. Consider the angle at the vertex of the flat triangle corresponding to p. The limit of this angle when t goes to zero is defined as the Alexandrov angle of the two rays at p. (By definition of a CAT(0) space, the angle monotonically decreases as t decreases, so the limit exists.) Now we define ${\displaystyle \angle _{p}(\xi _{1},\xi _{2})}$ to be this angle.

To define the angular metric on the boundary ∂X that does not depend on the choice of p, we take the

over all points in X

${\displaystyle \angle (\xi _{1},\xi _{2}):=\sup _{p\in X}\angle _{p}(\xi _{1},\xi _{2}).}$

The Tits metric d_T is the

of lengths of all the curves on the boundary that connect them measured in the angular metric. If there is no such curve with finite length, the Tits distance between the two points is defined as infinity.

The ideal boundary of X equipped with the Tits metric is called the Tits boundary, denoted as ∂_TX.

For a complete CAT(0) space, it can be shown that its ideal boundary with the angular metric is a complete CAT(1) space, and its Tits boundary is also a complete CAT(1) space. Thus for any two boundary points ${\displaystyle \xi _{1},\xi _{2}}$ such that ${\displaystyle \angle (\xi _{1},\xi _{2})<\pi }$, we have

${\displaystyle d_{\mathrm {T} }(\xi _{1},\xi _{2})=\angle (\xi _{1},\xi _{2}),}$

and the points can be joined by a unique geodesic segment on the boundary. If the space is

, then any two boundary points at finite Tits distance apart can be joined by a geodesic segment on the boundary.

Examples

• For a Euclidean space Eⁿ, its Tits boundary is the unit sphere S^{n - 1}.
• An Hadamard space X is called a visibility space if any two distinct boundary points are the endpoints of a geodesic line in X. For such a space, the angular distance between any two boundary points is equal to π, so there is no curve with finite length on the ideal boundary that connects any two distinct boundary points, which means that the Tits distance between any two of them is infinity.

References

• Bridson, Martin R.; Haefliger, André (1999). Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 319. Berlin: Springer-Verlag. pp. xxii+643.
  MR 1744486
  .