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Busemann function

In geometric topology, Busemann functions are used to study the large-scale geometry of geodesics in Hadamard spaces and in particular Hadamard manifolds (simply connected complete Riemannian manifolds of nonpositive curvature). They are named after Herbert Busemann, who introduced them; he gave an extensive treatment of the topic in his 1955 book "The geometry of geodesics".

Definition and elementary properties
Let $$(X,d)$$ be a metric space. A geodesic ray is a path $$\gamma : [0,\infty) \to X$$ which minimizes distance everywhere along its length. i.e., for all $$t,t' \in [0,\infty)$$, $$d\big(\gamma(t), \gamma(t') \big) = \big| t - t' \big|.$$ Equivalently, a ray is an isometry from the "canonical ray" (the set $$[0,\infty)$$ equipped with the Euclidean metric) into the metric space X.

Given a ray γ, the Busemann function $$B_\gamma : X \to \mathbb R$$ is defined by

$$B_\gamma(x)=\lim_{t\to\infty}\big(d\big( \gamma(t), x \big) - t \big)$$

Thus, when t is very large, the distance $$d\big( \gamma(t), x \big)$$ is approximately equal to $$ B_\gamma(x) + t $$. Given a ray γ, its Busemann function is always well-defined: indeed the right hand side above $$F_t(x) \stackrel{\text{def}}{=} d\big( \gamma(t), x \big) - t  $$, tends pointwise to the left hand side on compacta, since $$t-d( \gamma(t), x )=d(\gamma(t),\gamma(0)) - d(\gamma(t),x)$$ is bounded above by $$ d(\gamma(0),x)$$ and non-increasing since, if $$s\le t$$,

$$t-s + d(x,\gamma(s)) - d(x,\gamma(t)) \ge t -s -d (\gamma(s),\gamma(t)) = 0.$$

It is immediate from the triangle inequality that

$$|B_\gamma(x) - B_\gamma(y)|\le d(x,y),$$

so that $$B_\gamma$$ is uniformly continuous. More specifically, the above estimate above shows that

 * Busemann functions are Lipschitz functions with constant 1.

By Dini's theorem, the functions $$F_t(x)=d(x,\gamma(t)) -t $$ tend to $$B_\gamma(x)$$ uniformly on compact sets as t tends to infinity.

Example: Poincaré disk
Let $$D$$ be the unit disk in the complex plane with the Poincaré metric

$$ ds^2 = {4 \,|dz|^2 \over (1 - |z|^2)^2}.$$

Then, for $$|z| < 1$$ and $$|\zeta| = 1$$, the Busemann function is given by

$$ B_\zeta(z) = - \log\left({1 - |z|^2\over |z - \zeta|^2}\right),$$

where the term in brackets on the right hand side is the Poisson kernel for the unit disk and $$\zeta$$ corresponds to the radial geodesic $$\gamma$$ from the origin towards $$\zeta$$, $$\gamma(t) = \zeta \tanh(t/2)$$. The computation of $$d(x,y)$$ can be reduced to that of $$d(z,0) = d(|z|,0) = 2 \operatorname{artanh}(|z|) = \log \left( \tfrac{1+|z|}{1-|z|} \right) $$, since the metric is invariant under Möbius transformations in $$SU(1,1)$$; the geodesics through $$0$$ have the form $$\zeta g_t(0)$$ where $$g_t$$ is the 1-parameter subgroup of $$SU(1,1)$$,

$$g_t=\begin{pmatrix}\cosh (t/2) & \sinh (t/2) \\ \sinh (t/2) & \cosh (t/2)\end{pmatrix}$$

The formula above also completely determines the Busemann function by Möbius invariance.

Busemann functions on a Hadamard space
In a Hadamard space, where any two points are joined by a unique geodesic segment, the function $$F = F_t$$ is convex, i.e. convex on geodesic segments $$[x,y]$$. Explicitly this means that if $$z(s)$$ is the point which divides $$[x,y]$$ in the ratio $s : (1 − s)$, then $$F(z(s)) \leq s F(x) + (1 - s) F(y)$$. For fixed $$a$$ the function $$d(x,a)$$ is convex and hence so are its translates; in particular, if $$\gamma$$ is a geodesic ray in $$X$$, then $$F_t$$ is convex. Since the Busemann function $$B_\gamma$$ is the pointwise limit of $$F_t$$,

 * Busemann functions are convex on Hadamard spaces.
 * On a Hadamard space, the functions $$F_t(y)=d(y,\gamma(t)) -t $$ converge uniformly to $$B_\gamma$$ uniformly on any bounded subset of $$X$$.

Let $h(t) = d(y,γ(t)) − t = F_{t}(y)$. Since $$\gamma (t)$$ is parametrised by arclength, Alexandrov's first comparison theorem for Hadamard spaces implies that the function $g(t) = d(y,γ(t))^{2} − t^{2}$ is convex. Hence for $0< s < t$

$$g(s) \le (1-{s\over t})g(0) + {s\over t} g(t).$$

Thus

$$2sh(s)\le (h(s) +s)^2 - s^2 =g(s) \le (1-{s\over t})d(x,y)^2 + {s\over t} ( 2th(t) +h(t)^2 )\le d(x,y)^2 + 2s h(t) +{s\over t} d(x,y)^2,$$

so that

$$|F_s(y) - F_t(y)|= |h(s) - h(t)| \le {1\over 2} (s^{-1} + t^{-1}) d(x,y)^2.$$

Letting t tend to ∞, it follows that

$$|F_s(y) - B_\gamma(y)| \le {d(x,y)^2\over 2s},$$

so convergence is uniform on bounded sets.

Note that the inequality above for $$|F_s(y) - F_t(y)| $$ (together with its proof) also holds for geodesic segments: if $γ(t)$ is a geodesic segment starting at $x$ and parametrised by arclength then

$$|d(y,\Gamma(s)) -s -d(y,\Gamma(t)) +t| \le (s^{-1} + t^{-1}) d(x,y)^2.$$

Next suppose that $x, y$ are points in a Hadamard space, and let $δ(s)$ be the geodesic through $x$ with $δ(0) = y$ and $δ(t) = x$, where $t = d(x,y)$. This geodesic cuts the boundary of the closed ball $\overline{B}(y,r)$ at the point $δ(r)$. Thus if $d(x,y) > r$, there is a point $v$ with $d(y,v) = r$ such that $d(x,v) = d(x,y) − r$.

This condition persists for Busemann functions. The statement and proof of the property for Busemann functions relies on a fundamental theorem on closed convex subsets of a Hadamard space, which generalises orthogonal projection in a Hilbert space: if $C$ is a closed convex set in a Hadamard space $X$, then every point $x$ in $X$ has a unique closest point $P(x) ≡ P_{C}(x)$ in $C$ and $d(P(x),P(y)) ≤ d(x,y)$; moreover $a = P(x)$ is uniquely determined by the property that, for $y$ in $C$,

$$d(x,y)^2 \ge d(x,a) ^2 + d(a,y)^2,$$

so that the angle at $a$ in the Euclidean comparison triangle for $a,x,y$ is greater than or equal to $π/2$.

 * If $h$ is a Busemann function on a Hadamard space, then, given $y$ in $X$ and $r > 0$, there is a unique point $v$ with $d(y,v) = r$ such that $h(v) = h(y) − r$. For fixed $r > 0$, the point $v$ is the closest point of $y$ to the closed convex $C$ set of points $u$ such that $h(u) ≤ h(y) − r$ and therefore depends continuously on $y$.

Let $v$ be the closest point to $y$ in $C$. Then $h(v) = h(y) − r$ and so $h$ is minimised by $v$ in $\overline{B}(y,R)$ where $R = d(y,v) and v$ is the unique point where $h$ is minimised. By the Lipschitz condition $r = |h(y) − h(v)| ≤ R$. To prove the assertion, it suffices to show that $R = r$, i.e. $d(y,v) = r$. On the other hand, $h$ is the uniform limit on any closed ball of functions $h_{n}$. On $\overline{B}(y,r)$, these are minimised by points $v_{n}$ with $h_{n}(v_{n}) = h_{n}(y) − r$. Hence the infimum of $h$ on $\overline{B}(y,r)$ is $h(y) − r$ and $h(v_{n})$ tends to $h(y) − r$. Thus $h(v_{n}) = h(y) − r_{n}$ with $r_{n} ≤ r$ and $r_{n}$ tending towards $r$. Let $u_{n}$ be the closest point to $y$ with $h(u_{n}) ≤ h(y) − r_{n}$. Let $R_{n} = d(y,u_{n}) ≤ r$. Then $h(u_{n}) = h(y) − r_{n}$, and, by the Lipschitz condition on $h$, $R_{n} ≥ r_{n}$. In particular $R_{n}$ tends to $r$. Passing to a subsequence if necessary it can be assumed that $r_{n}$ and $R_{n}$ are both increasing (to $r$). The inequality for convex optimisation implies that for $n > m$.

$$ d(u_n,u_m)^2 \le R_n^2 - R_m^2\le 2r|R_n -R_m|,$$

so that $u_{n}$ is a Cauchy sequence. If $u$ is its limit, then $d(y,u) = r$ and $h(u) = h(y) − r$. By uniqueness it follows that $u = v$ and hence $d(y,v) = r$, as required.

Uniform limits. The above argument proves more generally that if $d(x_{n},x_{0})$ tends to infinity and the functions $h_{n}(x) = d(x,x_{n}) – d(x_{n},x_{0})$ tend uniformly on bounded sets to $h(x)$, then $h$ is convex, Lipschitz with Lipschitz constant 1 and, given $y$ in $X$ and $r > 0$, there is a unique point $v$ with $d(y,v) = r$ such that $h(v) = h(y) − r$. If on the other hand the sequence $(x_{n})$ is bounded, then the terms all lie in some closed ball and uniform convergence there implies that $(x_{n})$ is a Cauchy sequence so converges to some $x_{∞}$ in $X$. So $h_{n}$ tends uniformly to $h_{∞}(x) = d(x,x_{∞}) – d(x_{∞},x_{0})$, a function of the same form. The same argument also shows that the class of functions which satisfy the same three conditions (being convex, Lipschitz and having minima on closed balls) is closed under taking uniform limits on bounded sets.

Comment. Note that, since any closed convex subset of a Hadamard subset of a Hadamard space is also a Hadamard space, any closed ball in a Hadamard space is a Hadamard space. In particular it need not be the case that every geodesic segment is contained in a geodesic defined on the whole of $R$ or even a semi-infinite interval $[0,∞)$. The closed unit ball of a Hilbert space gives an explicit example which is not a proper metric space.

 * If $h$ is a convex function, Lipschitz with constant 1 and $h$ assumes its minimum on any closed ball centred on $y$ with radius $r$ at a unique point $v$ on the boundary with $h(v) = h(y) − r$, then for each $y$ in $X$ there is a unique geodesic ray $δ$ such that $δ(0) = y$ and $δ$ cuts each closed convex set $h ≤ h(y) – r$ with $r > 0$ at $δ(r)$, so that $h(δ(t)) = h(y) – t$. In particular this holds for each Busemann function.

The third condition implies that $v$ is the closest point to $y$ in the closed convex set $C_{r}$ of points $$u$$ such that $h(u) ≤ h(y) – r$. Let $δ(t)$ for $0 ≤ t ≤ r$ be the geodesic joining $y$ to $v$. Then $k(t) = h(δ(t)) - h(y)$ is a convex Lipschitz function on $[0,r]$ with Lipschitz constant 1 satisfying $k(t) ≤ – t$ and $k(0) = 0$ and $k(r) = –r$. So $k$ vanishes everywhere, since if $0 < s < r, k(s) ≤ –s$ and $|k(s)| ≤ s$. Hence $h(δ(t)) = h(y) – t$. By uniqueness it follows that $δ(t)$ is the closest point to $y$ in $C_{t}$ and that it is the unique point minimising $h$ in $\overline{B}(y,t)$. Uniqueness implies that these geodesics segments coincide for arbitrary $r$ and therefore that $δ$ extends to a geodesic ray with the stated property.

 * If $h = h_{γ}$, then the geodesic ray $δ$ starting at $y$ satisfies $$\sup d(\gamma(t),\delta(t)) < \infty$$. If $δ_{1}$ is another ray starting at $y$ with $$\sup d(\delta(t),\delta_1(t)) < \infty$$ then $δ_{1} = δ$.

To prove the first assertion, it is enough to check this for $t$ sufficiently large. In that case $γ(t)$ and $δ(t − h(y))$ are the projections of $x$ and $y$ onto the closed convex set $h ≤ −t$. Therefore, $d(γ(t),δ(t − h(y))) ≤ d(x,y)$. Hence $d(γ(t),δ(t)) ≤ d(γ(t),δ(t − h(y))) + d(δ(t − h(y)),δ(t)) ≤ d(x,y) + |h(y)|$. The second assertion follows because $d(δ_{1}(t),δ(t))$ is convex and bounded on $[0,∞)$, so, if it vanishes at $t = 0$, must vanish everywhere.

 * Suppose that $h$ is a continuous convex function and for each $y$ in $X$ there is a unique geodesic ray $δ$ such that $δ(0) = y$ and $δ$ cuts each closed convex set $h ≤ h(y) – r with r > 0$ at $δ(r)$, so that $h(δ(t)) = h(y) – t$; then $h$ is a Busemann function. $h − h_{δ}$ is a constant function.

Let $C_{r}$ be the closed convex set of points $z$ with $h(z) ≤ −r$. Since $X$ is a Hadamard space for every point $y$ in $X$ there is a unique closest point $P_{r}(y)$ to $y$ in $C_{r}$. It depends continuously on $y$ and if $y$ lies outside $C_{r}$, then $P_{r}(y)$ lies on the hypersurface $h(z) = − r$—the boundary ∂$C_{r}$ of $C_{r}$—and $P_{r}(y)$ satisfies the inequality of convex optimisation. Let $δ(s)$ be the geodesic ray starting at $y$.

Fix $x$ in $X$. Let $γ(s)$ be the geodesic ray starting at $x$. Let $g(z) = h_{γ}(z)$, the Busemann function for $γ$ with base point $x$. In particular $g(x) = 0$. It suffices to show that $g = h – h(y)1$. Now take $y$ with $h(x) = h(y)$ and let $δ(t)$ be the geodesic ray starting at $y$ corresponding to $h$. Then

$$d(x,y) \ge d(\gamma(t),\delta(t)),\,\,\, d(x,\delta(t))^2 \ge d(x,\gamma(t))^2+d(\gamma(t),\delta(t))^2,\,\,\,d(y,\gamma(t))^2 \ge d(y,\delta(t))^2+d(\gamma(t),\delta(t))^2.$$

On the other hand, for any four points $a$, $b$, $c$, $d$ in a Hadamard space, the following quadrilateral inequality of Reshetnyak holds:

$$ |d(a,c)^2 + d(b,d)^2 - d(a,d)^2 - d(b,d)^2|\le 2 d(a,b)d(c,d).$$

Setting $a = x$, $b = y$, $c = γ(t)$, $d = δ(t)$, it follows that

$$|d(y,\gamma(t))^2 - d(x,\delta(t))^2|\le 2d(x,y)^2,$$

so that

$$|d(y,\gamma(t)) - d(x,\gamma(t))|\le 2 {d(x,y)^2\over d(y,\gamma(t)) + d(x,\gamma(t))}\le {d(x,y)^2\over t}.$$

Hence $h_{γ}(y) = 0$. Similarly $h_{δ}(x) = 0$. Hence $h_{γ}(y) = 0$ on the level surface of $h$ containing $x$. Now for $t ≥ 0$ and $z$ in $X$, let $α_{t}(z) = γ_{1}(t)$ the geodesic ray starting at $z$. Then $α_{s + t} = α_{s} ∘ α_{t}$ and $h ∘ α_{t} = h − t$. Moreover, by boundedness, $d(α_{t}(u),α_{t}(v)) ≤ d(u,v)$. The flow $α_{s}$ can be used to transport this result to all the level surfaces of $h$. For general $y_{1}$, if $h(y_{1}) < h(x)$, take $s > 0$ such that $h(α_{s}(x)) = h(y_{1})$ and set $x_{1} = α_{s}(x)$. Then $h_{γ_{1}}(y_{1}) = 0|undefined$, where $γ_{1}(t) = α_{t}(x_{1}) = γ(s + t)$. But then $h_{γ_{1}} = h_{γ} – s|undefined$, so that $h_{γ}(y_{1}) = s$. Hence $g(y_{1}) = s = h((α_{s}(x)) – h(x) = h(y_{1}) – h(x)$, as required. Similarly if $h(y_{1}) > h(x)$, take $s > 0$ such that $h(α_{s}(y_{1})) = h(x)$. Let $y = α_{s}(y_{1})$. Then $h_{γ}(y) = 0$, so $h_{γ}(y_{1}) = –s$. Hence $g(y_{1}) = –s = h(y_{1}) – h(x)$, as required.

Finally there are necessary and sufficient conditions for two geodesics to define the same Busemann function up to constant:

 * On a Hadamard space, the Busemann functions of two geodesic rays $$\gamma_1$$ and $$\gamma_2$$ differ by a constant if and only if $$\sup_{t\ge 0} d(\gamma_1(t),\gamma_2(t)) < \infty$$.

Suppose firstly that $γ$ and $δ$ are two geodesic rays with Busemann functions differing by a constant. Shifting the argument of one of the geodesics by a constant, it may be assumed that $B_{γ} = B_{δ} = B$, say. Let $C$ be the closed convex set on which $B(x) ≤ −r$. Then $B(γ(t)) = B_{γ}(γ(t)) = −t$ and similarly $B(δ(t)) = − t$. Then for $s ≤ r$, the points $γ(s)$ and $δ(s)$ have closest points $γ(r)$ and  $δ(r)$ in $C$, so that $d(γ(r), δ(r)) ≤ d(γ(s), δ(s))$. Hence $sup_{t ≥ 0} d(γ(t), δ(t)) < ∞$.

Now suppose that $sup_{t ≥ 0} d(γ_{1}(t), γ_{2}(t)) < ∞$. Let $δ_{i}(t)$ be the geodesic ray starting at $y$ associated with $h_{γ_{i}}|undefined$. Then $sup_{t ≥ 0} d(γ_{i}(t), δ_{i}(t)) < ∞$. Hence $sup_{t ≥ 0} d(δ_{1}(t), δ_{2}(t)) < ∞$. Since $δ_{1}$ and $δ_{2}$ both start at $y$, it follows that $δ_{1}(t) ≡ δ_{2}(t)$. By the previous result $h_{γ_{i}}|undefined$ and $h_{δ_{i}}|undefined$ differ by a constant; so $h_{γ_{1}}|undefined$ and $h_{γ_{2}}|undefined$ differ by a constant.

To summarise, the above results give the following characterisation of Busemann functions on a Hadamard space:

THEOREM. On a Hadamard space, the following conditions on a function $f$ are equivalent:

 * $h$ is a Busemann function.
 * $h$ is a convex function, Lipschitz with constant $1$ and $h$ assumes its minimum on any closed ball centred on $y$ with radius $r$ at a unique point $v$ on the boundary with $h(v) = h(y) − r$.
 * $h$ is a continuous convex function and for each $y$ in $X$ there is a unique geodesic ray $δ$ such that $δ(0) = y$ and, for any $r > 0$, the ray $δ$ cuts each closed convex set $h ≤ h(y) – r$ at $δ(r)$.

Bordification of a Hadamard space
In the previous section it was shown that if $X$ is a Hadamard space and $x_{0}$ is a fixed point in $X$ then the union of the space of Busemann functions vanishing at $x_{0}$ and the space of functions $h_{y}(x) = d(x,y) − d(x_{0},y)$ is closed under taking uniform limits on bounded sets. This result can be formalised in the notion of bordification of $X$. In this topology, the points $x_{n}$ tend to a geodesic ray $γ$ starting at $x_{0}$ if and only if $d(x_{0},x_{n})$ tends to $∞$ and for $t > 0$ arbitrarily large the sequence obtained by taking the point on each segment $[x_{0},x_{n}]$ at a distance $t$ from $x_{0}$ tends to $γ(t)$.

If $X$ is a metric space, Gromov's bordification can be defined as follows. Fix a point $x_{0}$ in $X$ and let $X_{N} = \overline{B}(x_{0},N)$. Let $Y = C(X)$ be the space of Lipschitz continuous functions on $X$, i.e. those for which $| f(x) – f(y) | ≤ A d(x,y)$ for some constant $A > 0$. The space $Y$ can be topologised by the seminorms $\| f \|_{N} = sup_{X_{N}} | f ||undefined$, the topology of uniform convergence on bounded sets. The seminorms are finite by the Lipschitz conditions. This is the topology induced by the natural map of $C(X)$ into the direct product of the Banach spaces $C_{b}(X_{N})$ of continuous bounded functions on $X_{N}$. It is give by the metric $D(f,g) = Σ 2^{−N} \| f − g \|_{N}(1 +\| f − g \|_{N})^{−1}$.

The space $X$ is embedded into $Y$ by sending $x$ to the function $f_{x}(y) = d(y,x) – d(x_{0},x)$. Let $\overline{X}$ be the closure of $X$ in $Y$. Then $\overline{X}$ is metrisable, since $Y$ is, and contains $X$ as an open subset; moreover bordifications arising from different choices of basepoint are naturally homeomorphic. Let $h(x) = (d(x,x_{0}) + 1)^{−1}$. Then $h$ lies in $C_{0}(X)$. It is non-zero on $X$ and vanishes only at $∞$. Hence it extends to a continuous function on $\overline{X}$ with zero set $\overline{X} \ X$. It follows that $\overline{X} \ X$ is closed in $\overline{X}$, as required. To check that $\overline{X} = \overline{X}(x_{0})$ is independent of the basepoint, it suffices to show that $k(x) = d(x,x_{0}) − d(x,x_{1})$ extends to a continuous function on $\overline{X}$. But $k(x) = f_{x}(x_{1})$, so, for $g$ in $\overline{X}$, $k(g) = g(x_{1})$. Hence the correspondence between the compactifications for $x_{0}$ and $x_{1}$ is given by sending $g$ in $\overline{X}(x_{0})$ to $g + g(x_{1})1$ in $\overline{X}(x_{1})$.

When $X$ is a Hadamard space, Gromov's ideal boundary $∂X = \overline{X} \ X$ can be realised explicitly as "asymptotic limits" of geodesic rays using Busemann functions. If $x_{n}$ is an unbounded sequence in $X$ with $h_{n}(x) = d(x,x_{n}) − d(x_{n},x_{0})$ tending to $h$ in $Y$, then $h$ vanishes at $x_{0}$, is convex, Lipschitz with Lipschitz constant $1$ and has minimum $h(y) − r$ on any closed ball $\overline{B}(y,r)$. Hence $h$ is a Busemann function $B_{γ}$ corresponding to a unique geodesic ray $γ$ starting at $x_{0}$.

On the other hand, $h_{n}$ tends to $B_{γ}$ uniformly on bounded sets if and only if $d(x_{0},x_{n})$ tends to $∞$ and for $t > 0$ arbitrarily large the sequence obtained by taking the point on each segment $[x_{0},x_{n}]$ at a distance $t$ from $x_{0}$ tends to $γ(t)$. For $d(x_{0},x_{n}) ≥ t$, let $x_{n}(t)$ be the point in $[x_{0},x_{n}]$ with $d(x_{0},x_{n}(t)) = t$. Suppose first that $h_{n}$ tends to $B_{γ}$ uniformly on $\overline{B}(x_{0},R)$. Then for $t ≤ R$, $|h_{n}(γ(t)) – B_{γ}(γ(t))|=d(γ(t),x_{n}) – d(x_{n},x_{0}) + t$. This is a convex function. It vanishes as $t = 0$ and hence is increasing. So it is maximised at $t = R$. So for each $t$, $|d(γ(t),x_{n}) – d(x_{n},x_{0}) – t|$ tends towards 0. Let $a = X_{0}$, $b = γ(t)$ and $c = x_{n}$. Then $d(c,a) – d(c,b)$ is close to $d(a,b)$ with $d(c,a)$ large. Hence in the Euclidean comparison triangle $CA - CB$ is close to $AB$ with $CA$ large. So the angle at $A$ is small. So the point $D$ on $AC$ at the same distance as $AB$ lies close to $B$. Hence, by the first comparison theorem for geodesic triangles, $d(x_{n}(t),γ(t))$ is small. Conversely suppose that for fixed $t$ and $n$ sufficiently large $d(x_{n}(t),γ(t))$ tends to 0. Then from the above $F_{s}(y) = d(y,γ(s)) – s$ satisfies

$$|F_s(y) - B_\gamma(y)| \le {d(x_0,y)^2\over 2s},$$

so it suffices show that on any bounded set $h_{n}(y) = d(y,x_{n}) – d(x_{0},x_{n})$ is uniformly close to $F_{s}(y)$ for $n$ sufficiently large.

For a fixed ball $\overline{B}(x_{0},R)$, fix $s$ so that $R^{2}/s ≤ ε$. The claim is then an immediate consequence of the inequality for geodesic segments in a Hadamard space, since

$$|d(y,x_n) -d(y,x_0) - d(y,x_n(s)) + s |\le {d(x_0,y)^2\over s} \le \varepsilon.$$

Hence, if $y$ in $\overline{B}(x_{0},R)$ and $n$ is sufficiently large that $d(x_{n}(s),γ(s)) ≤ ε$, then

$$|h_n(y)-B_\gamma(y)|=|d(y,x_n) -d(y,x_0) - B_\gamma(y)| \le |d(y,x_n) -d(y,x_0) - d(y,x_n(s)) + s | + d(x_n(s),\gamma(s)) + |F_s(y) - B_\gamma(y)| \le 3\varepsilon.$$

Busemann functions on a Hadamard manifold
Suppose that $x, y$ are points in a Hadamard manifold and let $γ(s)$ be the geodesic through $x$ with $γ(0) = y$. This geodesic cuts the boundary of the closed ball $\overline{B}(y,r)$ at the two points $γ(±r)$. Thus if $d(x,y) > r$, there are points $u, v$ with $d(y,u) = d(y,v) = r$ such that $| d(x,u) − d(x,v) | = 2r$. By continuity this condition persists for Busemann functions:

 * If $h$ is a Busemann function on a Hadamard manifold, then, given $y$ in $X$ and $r > 0$, there are unique points $u$, $v$ with $d(y,u) = d(y,v) = r$ such that $h(u) = h(y) + r$ and $h(v) = h(y) − r$. For fixed $r > 0$, the points $u$ and $v$ depend continuously on $y$.

Taking a sequence $t_{n}$ tending to $∞$ and $h_{n} = F_{t_{n}}|undefined$, there are points $u_{n}$ and $v_{n}$ which satisfy these conditions for $h_{n}$ for $n$ sufficiently large. Passing to a subsequence if necessary, it can be assumed that $u_{n}$ and $v_{n}$ tend to $u$ and $v$. By continuity these points satisfy the conditions for $h$. To prove uniqueness, note that by compactness $h$ assumes its maximum and minimum on $\overline{B}(y,r)$. The Lipschitz condition shows that the values of $h$ there differ by at most $2r$. Hence $h$ is minimized at $v$ and maximized at $u$. On the other hand, $d(u,v) = 2r$ and for $u$ and $v$ the points $v$ and $u$ are the unique points in $\overline{B}(y,r)$ maximizing this distance. The Lipschitz condition on $h$ then immediately implies $u$ and $v$ must be the unique points in $\overline{B}(y,r)$ maximizing and minimizing $h$. Now suppose that $y_{n}$ tends to $y$. Then the corresponding points $u_{n}$ and $v_{n}$ lie in a closed ball so admit convergent subsequences. But by uniqueness of $u$ and $v$ any such subsequences must tend to $u$ and $v$, so that $u_{n}$ and $v_{n}$ must tend to $u$ and $v$, establishing continuity.

The above result holds more generally in a Hadamard space.

 * If $h$ is a Busemann function on a Hadamard manifold, then $h$ is continuously differentiable with $\| dh(y) \| = 1$ for all $y$.

From the previous properties of $h$, for each $y$ there is a unique geodesic γ(t) parametrised by arclength with $γ(0) = y$ such that $h ∘ γ(t) = h(y) + t$. It has the property that it cuts $∂B(y,r)$ at $t = ±r$: in the previous notation $γ(r) = u$ and $γ(–r) = v$. The vector field $V_{h}$ defined by the unit vector $$\dot{\gamma}(0)$$ at $y$ is continuous, because $u$ is a continuous function of $y$ and the map sending $(x,v)$  to $(x,exp_{x} v)$ is a diffeomorphism from $TX$ onto $X × X$ by the Cartan-Hadamard theorem. Let $δ(s)$ be another geodesic parametrised by arclength through $y$ with $δ(0) = y$. Then $dh ∘ δ (0)/ ds =$ $$(\dot{\delta}(0),\dot{\gamma}(0))$$. Indeed, let $H(x) = h(x) − h(y)$, so that $H(y) = 0$. Then

$$ |H(\delta(s)) - H(x)| \le d(\delta(s),x).$$

Applying this with $x = u$ and $v$, it follows that for $s > 0$

$$(r- d(\delta(s),u))/s \le (h(\delta(s)) - h(y))/s \le (d(\delta(s),v) -r)/s.$$

The outer terms tend to $$(\dot{\delta}(0),\dot{\gamma}(0))$$ as $s$ tends to 0, so the middle term has the same limit, as claimed. A similar argument applies for $s < 0$.

The assertion on the outer terms follows from the first variation formula for arclength, but can be deduced directly as follows. Let $$a=\dot{\delta}(0)$$ and $$b=\dot{\gamma}(0)$$, both unit vectors. Then for tangent vectors $p$ and $q$ at $y$ in the unit ball

$$d(\exp_y p,\exp_y q) = \|p-q\| + \varepsilon \max \|p\|^2, \|q\|^2$$ with $g(x) = I + ε \| x \|$ uniformly bounded. Let $p$ and $q$. Then

$$(d(\delta(s),v) -r)/s= (d(\exp_y(t^3 a),\exp_y(-t^2 b)) - t^2)/t^3 = (\|t^3a +t^2b\| - t^2)/t^3 + \varepsilon |t|= (\|ta +b\| -1)/t + \varepsilon |t|.$$

The right hand side here tends to $r = max \| p \|, \| q \|$ as $d(p,q)$ tends to 0 since

$$ {d\over dt} \|b + t a\||_{t=0 }= {1\over 2} \,{d\over dt} \|b + t a\|^2|_{t=0} = (a,b).$$

The same method works for the other terms.

Hence it follows that $c(t)$ is a $p$ function with $q$ dual to the vector field $L(c) ≥ (1 − ε r) ⋅ ∫ \| c \| dt ≥ (1 − ε r) ⋅ \| p − q \|$, so that $g(x) = I + ε \|x\|^{2}$. The vector field $ε$ is thus the gradient vector field for $s = t^{3}$. The geodesics through any point are the flow lines for the flow $r = t^{2}$ for $(a,b)$, so that $t$ is the gradient flow for $h$.

THEOREM. On a Hadamard manifold $C^{1}$ the following conditions on a continuous function $dh$ are equivalent:
 * 1) $V_{h}$ is a Busemann function.
 * 2) $\| dh(y) \| = 1$ is a convex, Lipschitz function with constant 1, and for each $V_{h}$ in $h$ there are points $α_{t}$ at a distance $V_{h}$ from $α_{t}$ such that $h$.
 * 3) $X$ is a convex $h$ function with $h$.

It has already been proved that (1) implies (2).

The arguments above show mutatis mutandi that (2) implies (3).

It therefore remains to show that (3) implies (1). Fix $h$ in $y$. Let $X$ be the gradient flow for $u_{±}$. It follows that $r$ and that $y$ is a geodesic through $h(u_{±}) = h(y) ± r$ parametrised by arclength with $h$. Indeed, if $C^{1}$, then

$$|s-t|=|h(\alpha_s(x))-h(\alpha_t(x))| \le d(\alpha_s(x),\alpha_t(x))\le \int_{s}^t \|d\alpha_\tau(x)/d\tau\| \, d\tau =\int_s^t \|dh(\alpha_\tau(x))\| \, d\tau = |s-t|,$$

so that $\| dh(x) \| ≡ 1$. Let $x$, the Busemann function for $X$ with base point $α_{t}$. In particular $h$. To prove (1), it suffices to show that $h ∘ α_{t} (x) = h(x) + t$.

Let $γ(t) = α_{t}(x)$ be the closed convex set of points $x$ with $γ(0) = x$. Since $s < t$ is a Hadamard space for every point $d(γ(s),γ(t)) = | s − t |$ in $g(y) = h_{γ}(y)$ there is a unique closest point $γ$ to $x$ in $g(x) = 0$. It depends continuously on $g = h – h(x)1$ and if $C(−r)$ lies outside $z$, then $h(z) ≤ −r$ lies on the hypersurface $X$—the boundary $y$ of $X$—and the geodesic from $P_{r}(y)$ to $y$ is orthogonal to $C(-r)$. In this case the geodesic is just $y$. Indeed, the fact that $y$ is the gradient flow of $C(-r)$ and the conditions $P_{r}(y)$ imply that the flow lines $h(z) = − r$ are geodesics parametrised by arclength and cut the level curves of $∂C(–r)$ orthogonally. Taking $C(–r)$ with $y$ and $P_{r}(y)$,

$$d(x,y) \ge d(\alpha_t(x),\alpha_t(y)),\,\,\, d(x,\alpha_t(y))^2 \ge d(x,\alpha_t(x))^2+d(\alpha_t(x),\alpha_t(y))^2,\,\,\,d(y,\alpha_t(x))^2 \ge d(y,\alpha_t(y))^2+d(\alpha_t(x),\alpha_t(y))^2.$$

On the other hand, for any four points $∂C(–r)$, $α_{t}(y)$, $α_{t}$, $h$ in a Hadamard space, the following quadrilateral inequality of Reshetnyak holds:

$$ |d(a,c)^2 + d(b,d)^2 - d(a,d)^2 - d(b,d)^2|\le 2 d(a,b)d(c,d).$$

Setting $\| dh(y) \| ≡ 1$, $α_{t}(y)$, $h$, $y$, it follows that

$$|d(y,\alpha_t(x))^2 - d(x,\alpha_t(x))^2|\le 2d(x,y)^2,$$

so that

$$|d(y,\alpha_t(x)) - d(x,\alpha_t(x))|\le 2 {d(x,y)^2\over d(y,\alpha_t(x)) + d(x,\alpha_t(x))}\le {d(x,y)^2\over t}.$$

Hence $h(y) = h(x)$ on the level surface of $t > 0$ containing $a$. The flow $b$ can be used to transport this result to all the level surfaces of $c$. For general $d$ take $a = x$ such that $b = y$ and set $c = α_{t}(x)$. Then $d = α_{t}(y)$, where $h_{γ}(y) = 0$. But then $h$, so that $x$. Hence $α_{s}$, as required.

Note that this argument could be shortened using the fact that two Busemann functions $h$ and $y_{1}$ differ by a constant if and only if the corresponding geodesic rays satisfy $s$. Indeed, all the geodesics defined by the flow $h(α_{s}(x)) = h(y_{1})$ satisfy the latter condition, so differ by constants. Since along any of these geodesics $x_{1} = α_{s}(x)$ is linear with derivative 1, $h_{γ_{1}}(y_{1}) = 0|undefined$ must differ from these Busemann functions by constants.

Compactification of a proper Hadamard space
defined a compactification of a Hadamard manifold $γ_{1}(t) = α_{t}(x_{1}) = γ(s + t)$ which uses Busemann functions. Their construction, which can be extended more generally to proper (i.e. locally compact) Hadamard spaces, gives an explicit geometric realisation of a compactification defined by Gromov—by adding an "ideal boundary"—for the more general class of proper metric spaces $h_{γ_{1}} = h_{γ} – s|undefined$, those for which every closed ball is compact. Note that, since any Cauchy sequence is contained in a closed ball, any proper metric space is automatically complete. The ideal boundary is a special case of the ideal boundary for a metric space. In the case of Hadamard spaces, this agrees with the space of geodesic rays emanating from any fixed point described using Busemann functions in the bordification of the space.

If $h_{γ}(y_{1}) = s$ is a proper metric space, Gromov's compactification can be defined as follows. Fix a point $g(y_{1}) = s = h((α_{s}(x)) – h(x) = h(y_{1}) – h(x)$ in $h_{γ}$ and let $h_{δ}$. Let $sup_{t ≥ 0} d(γ(t),δ(t)) < ∞$ be the space of Lipschitz continuous functions on $α_{t}$, .e. those for which $h$ for some constant $h$. The space $X$ can be topologised by the seminorms $X$, the topology of uniform convergence on compacta. This is the topology induced by the natural map of C(X) into the direct product of the Banach spaces $X$. It is give by the metric $R$.

The space $d(x,y) = | x – y | /(1 + | x – y |)$ is embedded into $X$ by sending $x$ to the function $y$. Let $X$ be the closure of $x$ in $X$. Then $K = \overline{B}(x,r)$ is compact (metrisable) and contains $\overline{B}(x,R)$ as an open subset; moreover compactifications arising from different choices of basepoint are naturally homeomorphic. Compactness follows from the Arzelà–Ascoli theorem since the image in $R > r$ is equicontinuous and uniformly bounded in norm by $(x_{n})$. Let $d(x,x_{n})$ be a sequence in $r$ tending to $y_{n}$ in $x$. Then all but finitely many terms must lie outside $x_{n}$ since $d(x,y_{n}) = r$ is compact, so that any subsequence would converge to a point in $K$; so the sequence $(y_{n})$ must be unbounded in $(x_{n})$. Let $X$. Then $x_{0}$ lies in $X$. It is non-zero on $X_{N} = \overline{B}(x_{0},N)$ and vanishes only at $Y = C(X)$. Hence it extends to a continuous function on $X$ with zero set $| f(x) – f(y) | ≤ A d(x,y)$. It follows that $A > 0$ is closed in $Y$, as required. To check that the compactification $\| f \|_{N} = sup_{X_{N}} | f ||undefined$ is independent of the basepoint, it suffices to show that $C(X_{N})$ extends to a continuous function on $D(f,g) = Σ 2^{−N} \| f − g \| _{N}(1 + \| f − g \| _{N})^{−1}$. But $X$, so, for $Y$ in $x$, $f_{x}(y) = d(y,x) – d(x_{0},x)$. Hence the correspondence between the compactifications for $\overline{X}$ and $X$ is given by sending $Y$ in $\overline{X}$ to $X$ in $C(X_{N})$.

When $N$ is a Hadamard manifold (or more generally a proper Hadamard space), Gromov's ideal boundary $x_{n}$ can be realised explicitly as "asymptotic limits" of geodesics by using Busemann functions. Fixing a base point $X ⊂ \overline{X}$, there is a unique geodesic $y$ parametrised by arclength such that $\overline{X} \ X$ and $$\dot{\gamma}(0)$$ is a given unit vector. If $X_{N}$ is the corresponding Busemann function, then $X_{N}$ lies in $X_{N}$ and induces a homeomorphism of the unit $x_{n}$-sphere onto $X$, sending $$\dot{\gamma}(0)$$ to $h(x) = (d(x,x_{0}) + 1)^{−1}$.

Morse–Mostow lemma
In the case of spaces of negative curvature, such as the Poincaré disk, CAT(-1) and hyperbolic spaces, there is a metric structure on their Gromov boundary. This structure is preserved by the group of quasi-isometries which carry geodesics rays to quasigeodesic rays. Quasigeodesics were first studied for negatively curved surfaces—in particular the hyperbolic upper halfplane and unit disk—by Morse and generalised to negatively curved symmetric spaces by Mostow, for his work on the rigidity of discrete groups. The basic result is the Morse–Mostow lemma on the stability of geodesics.

By definition a quasigeodesic Γ defined on an interval $h$ with $C_{0}(X)$ is a map $X$ into a metric space, not necessarily continuous, for which there are constants $∞$ and $\overline{X}$ such that for all $\overline{X} \ X$ and $\overline{X} \ X$:

$$\lambda^{-1} |s-t| - \varepsilon \le d(\Gamma(s),\Gamma(t)) \le \lambda |s-t| + \varepsilon.$$

The following result is essentially due to Marston Morse (1924).

Morse's lemma on stability of geodesics. In the hyperbolic disk there is a constant $\overline{X}$ depending on $\overline{X} = \overline{X}(x_{0})$ and $k(x) = d(x,x_{0}) − d(x,x_{1})$ such that any quasigeodesic segment $\overline{X}$ defined on a finite interval $k(x) = f_{x}(x_{1})$ is within a Hausdorff distance $g$ of the geodesic segment $\overline{X}$.

Classical proof for Poincaré disk
The classical proof of Morse's lemma for the Poincaré unit disk or upper halfplane proceeds more directly by using orthogonal projection onto the geodesic segment.

 * It can be assumed that Γ satisfies the stronger "pseudo-geodesic" condition:

$$\lambda^{-1}|s-t| -\varepsilon \le d(\Gamma(s),\Gamma(t)) \le \lambda |s-t|.$$

$k(g) = g(x_{1})$ can be replaced by a continuous piecewise geodesic curve Δ with the same endpoints lying at a finite Hausdorff distance from $x_{0}$ less than $x_{1}$: break up the interval on which $g$ is defined into equal subintervals of length $\overline{X}(x_{0})$ and take the geodesics between the images under $g + g(x_{1})1$ of the endpoints of the subintervals. Since $\overline{X}(x_{1})$ is piecewise geodesic, $X$ is Lipschitz continuous with constant $∂X = \overline{X} \ X$, $x_{0}$, where $γ(t)$. The lower bound is automatic at the endpoints of intervals. By construction the other values differ from these by a uniformly bounded depending only on $γ(0) = x_{0}$ and $B_{γ}$; the lower bound inequality holds by increasing ε by adding on twice this uniform bound.

 * If $B_{γ}$ is a piecewise smooth curve segment lying outside an $∂X(x_{0})$-neighbourhood of a geodesic line and $(n − 1)$ is the orthogonal projection onto the geodesic line then:

$$\ell(P \circ \gamma) \le {\ell(\gamma)\over \cosh s}.$$

Applying an isometry in the upper half plane, it may be assumed that the geodesic line is the positive imaginary axis in which case the orthogonal projection onto it is given by $∂X(x_{0})$ and $B_{γ}$. Hence the hypothesis implies $[a,b]$, so that

$$ \ell(P\circ\gamma) = \int_a^b {|d\gamma|\over |\gamma|} \le \int_a^b {|d\gamma|\over \cosh(s)\, \Im \gamma}={\ell(\gamma)\over \cosh(s)}.$$

 * There is a constant $−∞ ≤ a < b ≤ ∞$ depending only on $Γ(t)$ and $λ ≥ 1$ such that $ε > 0$ lies within an $s$-neighbourhood of the geodesic segment $t$.

Let $R$ be the geodesic line containing the geodesic segment $λ$. Then there is a constant $ε$ depending only on $Γ$ and $[a,b]$ such that $R$-neighbourhood $[Γ(a),Γ(b)]$ lies within an $Γ$-neighbourhood of $Γ$. Indeed for any $c = (2λ^{2} + 1)ε$, the subset of $Γ$ for which $2λε$ lies outside the closure of the $Γ$-neighbourhood of $Δ$ is open, so a countable union of open intervals $Δ$. Then

$$\ell(\Gamma|_{[c,d]}) \le s_1 \equiv \lambda^2 (2s +\varepsilon) \left(1 -{\lambda^2\over \cosh(s)}\right),$$

 * since the left hand side is less than or equal to $λ_{1}$ and

$${|c-d|\over \lambda} -\varepsilon \le d(\Gamma(c),\Gamma(d)) \le 2s + d(P \circ \Gamma|_{[c,d]}) \le 2s + {\lambda |c-d|\over\cosh(s)}.$$

 * Hence every point lies at a distance less than or equal to $d(Δ(s),Δ(t)) ≤ λ_{1} | s – t |$ of $λ_{1} ≤ λ + ε$. To deduce the assertion, note that the subset of $λ$ for which $ε$ lies outside the closure of the $γ$-neighbourhood of $s$ is open, so a union of intervals $P$ with $P(z) = i | z |$ and $| z | / Im z = cosh d(z,Pz)$ both at a distance $| γ(t) | ≥ cosh(s) Im γ(t)$ from either $h > 0$ or $λ$. Then

$$\ell(\Gamma|_{[c,d]}) \le s_2 \equiv \lambda^2(2(s + s_1) + \varepsilon),$$

 * since

$${|c-d|\over \lambda} -\varepsilon \le d(\Gamma(c),\Gamma(d)) \le 2(s+s_1).$$

 * Hence the assertion follows taking any $ε$ greater than $Γ[a,b]$.

 * There is a constant $h$ depending only on $[Γ(a),Γ(b)]$ and $γ(t)$ such that the geodesic segment $[Γ(a),Γ(b)]$ lies within an $h > 0$-neighbourhood of $λ$.

Every point of $ε$ lies within a distance $h$ of $Γ[a,b]$. Thus orthogonal projection $h$ carries each point of $γ(R)$ onto a point in the closed convex set $s > 0$ at a distance less than $[a,b]$. Since $Γ(t)$ is continuous and $s$ connected, the map $γ(R)$ must be onto since the image contains the endpoints of $(c,d)$. But then every point of $λ | c – d |$ is within a distance $s + s_{1}$ of a point of $γ(R)$.

Gromov's proof for Poincaré disk
The generalisation of Morse's lemma to CAT(-1) spaces is often referred to as the Morse–Mostow lemma and can be proved by a straightforward generalisation of the classical proof. There is also a generalisation for the more general class of hyperbolic metric spaces due to Gromov. Gromov's proof is given below for the Poincaré unit disk; the properties of hyperbolic metric spaces are developed in the course of the proof, so that it applies mutatis mutandi to CAT(-1) or hyperbolic metric spaces.

Since this is a large-scale phenomenon, it is enough to check that any maps $[a,b]$ from $Γ(t)$ for any $s$ to the disk satisfying the inequalities is within a Hausdorff distance $[Γ(a),Γ(b)] ⊂ γ(R)$ of the geodesic segment $(c,d)$. For then translating it may be assumed without loss of generality $Γ(c)$ is defined on $Γ(d)$ with $s + s_{1}$ and then, taking $Γ(a)$ (the integer part of $Γ(b)$), the result can be applied to $h$ defined by $s +s_{1} + s_{2}$. The Hausdorff distance between the images of $h > 0$ and $λ$ is evidently bounded by a constant $ε$ depending only on $[Γ(a),Γ(b)]$ and $h$.

 * Now the incircle of a geodesic triangle has diameter less than $Γ[a,b]$ where $Γ[a,b]$; indeed it is strictly maximised by that of an ideal triangle where it equals $h$. In particular, since the incircle breaks the triangle breaks the triangle into three isosceles triangles with the third side opposite the vertex of the original triangle having length less than $[Γ(a),Γ(b)]$, it follows that every side of a geodesic triangle is contained in a $P$-neighbourhood of the other two sides. A simple induction argument shows that a geodesic polygon with $Γ[a,b]$ vertices for $[Γ(a),Γ(b)]$ has each side within a $h$ neighbourhood of the other sides (such a polygon is made by combining two geodesic polygons with $P$ sides along a common side). Hence if $Γ[a,b]$, the same estimate holds for a polygon with $P$ sides.

 * For $[Γ(a),Γ(b)]$ let $[Γ(a),Γ(b)]$, the largest radius for a closed ball centred on $h$ which contains no $Γ[a,b]$ in its interior. This is a continuous function non-zero on $Δ$ so attains its maximum $\{0, 1, 2, ..., N\}$ at some point $N > 0$ in this segment. Then $R_{1}$ lies within an $[Δ(0),Δ(N)]$-neighbourhood of the image of $Γ$ for any $[0,r]$. It therefore suffices to find an upper bound for $r > 1$ independent of $N = [r]$.

 * Choose $r$ and $Δ$ in the segment $Δ(i) = Γ(i)$ before and after $Γ$ with $Δ$ and $R_{2}$ (or an endpoint if it within a distance of less than $λ$ from $ε$). Then there are $δ$ with $δ = 2 log 3$, $2 log 3$. Hence $δ$, so that $δ$. By the triangle inequality all points on the segments $2^{k} + 2$ and $k ≥ 0$ are at a distance $(k + 1)δ$ from $2^{k−1} + 1$. Thus there is a finite sequence of points starting at $M ≤ 2^{k} + 2$ and ending at $M$, lying first on the segment $y_{i} = Δ(i)$, then proceeding through the points $f(x) = min d(x,y_{i})$, before taking the segment $x$. The successive points $y_{i}$ are separated by a distance no greater than $[Δ(0),Δ(N)]$ and successive points on the geodesic segments can also be chosen to satisfy this condition. The minimum number $h$ of points in such a sequence satisfies $x$. These points form a geodesic polygon, with $[Δ(0),Δ(N)]$ as one of the sides. Take $h_{1}$, so that the $Δ$-neighbourhood of $h_{1} > h$ does not contain all the other sides of the polygon. Hence, from the result above, it follows that $h$. Hence

$$3 + 2(\lambda+\varepsilon)^{-1}h + 6\lambda h + \varepsilon> 2^{h/\delta} + 2.$$

 * This inequality implies that $N$ is uniformly bounded, independently of $y$, as claimed.

 * If all points $z$ lie within $[Δ(0),Δ(N)]$ of the $x$, the result follows. Otherwise the points which do not fall into maximal subsets $d(x,y) = 2h$ with $d(x,z) = 2h$. Thus points in $2h$ have a point $x$ with $i, j$ in the complement of $d(y,Δ(i))$ within a distance of $d(z,Δ(j)) ≤ h$. But the complement of $d(Δ(i),Δ(j)) ≤ 6h$, a disjoint union with $| i − j | ≤ 6λh + λε$ and $[y,Δ(i)]$. Connectivity of $[z,Δ(j)]$ implies there is a point $≥ h$ in the segment which is within a distance $x$ of points $y$ and $z$ with $[y,Δ(i)]$ and $Δ(i), Δ(i+1), ..., Δ(j)$. But then $[Δ(j),z]$, so $Δ(i), Δ(i+1), ..., Δ(j)$. Hence the points $λ + ε$ for $K$ in $K ≤ | i - j | + 3 + 2(λ + ε)^{–1}h$ lie within a distance from  $[y,z]$ of less than $L = [h/δ]$.

Extension to quasigeodesic rays and lines
Recall that in a Hadamard space if $(L − 1)δ$ and $[y,z]$ are two geodesic segments and the intermediate points $K > 2^{L − 1} + 2$ and $h$ divide them in the ratio $N$, then $Δ(i)$ is a convex function of $h_{1}$. In particular if $[Δ(0),Δ(N)]$ and $S = \{r, ..., s\}$ are geodesic segments of unit speed defined on $r < s$ starting at the same point then

$$d(\Gamma_1(t),\Gamma_2(t)) \le {t\over R} d(\Gamma_1(R),\Gamma_2(R)).$$

In particular this implies the following:

 * In a CAT(–1) space $[Δ(0),Δ(N)]$, there is a constant $Δ(i)$ depending only on $i$ and $S$ such that any quasi-geodesic ray is within a bounded Hausdorff distance $h_{1}$ of a geodesic ray. A similar result holds for quasigeodesic and geodesic lines.

If $S = S_{1} ∐ S_{2}$ is a geodesic say with constant $S_{1} = \{0, ..., r − 1\}$ and $S_{2} = \{s + 1, ..., N\}$, let $[Δ(0),Δ(N)]$ be the unit speed geodesic for the segment $x$. The estimate above shows that for fixed $h_{1}$ and $Δ(i)$ sufficiently large, $Δ(j)$ is a Cauchy sequence in $i < r$ with the uniform metric. Thus $j > s$ tends to a geodesic ray $d(Δ(i),Δ(j)) < 2 h_{1}$ uniformly on compacta the bound on the Hausdorff distances between $| i − j | ≤ 2λh_{1} + λε$ and the segments $Δ(k)$ applies also to the limiting geodesic $k$. The assertion for quasigeodesic lines follows by taking $S$ corresponding to the geodesic segment $[Δ(0),Δ(N)]$.

Efremovich–Tikhomirova theorem
Before discussing CAT(-1) spaces, this section will describe the Efremovich–Tikhomirova theorem for the unit disk $h_{1} + λ | i − j | + ε ≤ h_{1} + λ (2λh_{1} + λε) + ε ≡ h_{2}$ with the Poincaré metric. It asserts that quasi-isometries of $[a_{1},b_{1}]$ extend to quasi-Möbius homeomorphisms of the unit disk with the Euclidean metric. The theorem forms the prototype for the more general theory of CAT(-1) spaces. Their original theorem was proved in a slightly less general and less precise form in and applied to bi-Lipschitz homeomorphisms of the unit disk for the Poincaré metric; earlier, in the posthumous paper, the Japanese mathematician Akira Mori had proved a related result within Teichmüller theory assuring that every quasiconformal homeomorphism of the disk is Hölder continuous and therefore extends continuously to a homeomorphism of the unit circle (it is known that this extension is quasi-Möbius).

Extension of quasi-isometries to boundary
If $[a_{2},b_{2}]$ is the Poincaré unit disk, or more generally a CAT(-1) space, the Morse lemma on stability of quasigeodesics implies that every quasi-isometry of $c_{1}(t)$ extends uniquely to the boundary. By definition two self-mappings $c_{2}(t)$ of $t:(1 – t)$ are quasi-equivalent if $d(c_{1}(t),c_{2}(t))$, so that corresponding points are at a uniformly bounded distance of each other. A quasi-isometry $t$ of $Γ_{1}(t)$ is a self-mapping of $Γ_{2}(t)$, not necessarily continuous, which has a quasi-inverse $[0,R]$ such that $X$ and $h > 0$ are quasi-equivalent to the appropriate identity maps and such that there are constants $λ$ and $ε$ such that for all $h$ in $Γ(t)$ and both mappings

$$ \lambda^{-1} d(x,y) - \varepsilon \le d(f_k(x),f_k(y)) \le \lambda d(x,y) +\varepsilon.$$

Note that quasi-inverses are unique up to quasi-equivalence; that equivalent definition could be given using possibly different right and left-quasi inverses, but they would necessarily be quasi-equivalent; that quasi-isometries are closed under composition which up to quasi-equivalence depends only the quasi-equivalence classes; and that, modulo quasi-equivalence, the quasi-isometries form a group.

Fixing a point $λ$ in $ε$, given a geodesic ray $Γ_{N}(t)$ starting at $[Γ(0),Γ(N)]$, the image $R > 0$ under a quasi-isometry $N$ is a quasi-geodesic ray. By the Morse-Mostow lemma it is within a bounded distance of a unique geodesic ray $(Γ_{N})$ starting at $C([0,R],X)$. This defines a mapping $Γ_{N}$ on the boundary $γ$ of $Γ$, independent of the quasi-equivalence class of $Γ_{N}$, such that $γ$. Thus there is a homomorphism of the group of quasi-isometries into the group of self-mappings of $Γ_{N}$.

To check that $[Γ(–N),Γ(N)]$ is continuous, note that if $D$ and $D$ are geodesic rays that are uniformly close on $X$, within a distance $X$, then $f, g$ and $X$ lie within a distance $sup_{X} d(f(x),g(x)) < ∞$ on $f_{1}$, so that $X$ and $X$ lie within a distance $f_{2}$; hence on a smaller interval $f_{1} ∘ f_{2}$, $f_{2} ∘ f_{1}$ and $λ ≥ 1$ lie within a distance $ε > 0$ by convexity.

On CAT(-1) spaces, a finer version of continuity asserts that $x, y$ is a quasi-Möbius mapping with respect to a natural class of metric on $X$, the "visual metrics" generalising the Euclidean metric on the unit circle and its transforms under the Möbius group. These visual metrics can be defined in terms of Busemann functions.

In the case of the unit disk, Teichmüller theory implies that the homomorphism carries quasiconformal homeomorphisms of the disk onto the group of quasi-Möbius homeomorphisms of the circle (using for example the Ahlfors–Beurling or Douady–Earle extension): it follows that the homomorphism from the quasi-isometry group into the quasi-Möbius group is surjective.

In the other direction, it is straightforward to prove that the homomorphism is injective. Suppose that $x$ is a quasi-isometry of the unit disk such that $X$ is the identity. The assumption and the Morse lemma implies that if $γ$ is a geodesic line, then $x$ lies in an $f ∘ γ$-neighbourhood of $f$. Now take a second geodesic line $δ$ such that $x$ and $∂f$ intersect orthogonally at a given point in $∂X$. Then $X$ lies in the intersection of $f$-neighbourhoods of $∂(f ∘ g) = ∂f ∘ ∂g$ and $∂X$. Applying a Möbius transformation, it can be assumed that $∂f$ is at the origin of the unit disk and the geodesics are the real and imaginary axes. By convexity, the $γ_{1}$-neighbourhoods of these axes intersect in a $γ_{2}$-neighbourhood of the origin: if $[0,R]$ lies in both neighbourhoods, let $η$ and $f ∘ γ_{1}$ be the orthogonal projections of $f ∘ γ_{2}$ onto the $λη + ε$- and $[0,R]$-axes; then $δ_{1}$ so taking projections onto the $δ_{2}$-axis, $λη + ε + 2h(λ,ε)$; hence $[0,r]$. Hence $δ_{1}$, so that $δ_{2}$ is quasi-equivalent to the identity, as claimed.

Cross ratio and distance between non-intersecting geodesic lines
Given two distinct points $(r/R)⋅[λη + ε + 2h(λ,ε)]$ on the unit circle or real axis there is a unique hyperbolic geodesic $∂f$ joining them. It is given by the circle (or straight line) which cuts the unit circle unit circle or real axis orthogonally at those two points. Given four distinct points $∂X$ in the extended complex plane their cross ratio is defined by

$$(a,b;c,d) ={(a-c)(b-d)\over (a-d)(b-c)}.$$

If $f$ is a complex Möbius transformation then it leaves the cross ratio invariant: $∂f$. Since the Möbius group acts simply transitively on triples of points, the cross ratio can alternatively be described as the complex number $γ(R)$ in $f(γ(R))$ such that $h$ for a Möbius transformation $γ(R)$.

Since $δ$, $δ$, $γ$ and $a$ all appear in the numerator defining the cross ratio, to understand the behaviour of the cross ratio under permutations of $f(a)$, $h$, $δ$ and $γ$, it suffices to consider permutations that fix $a$, so only permute $h$, $3h$ and $z$. The cross ratio transforms according to the anharmonic group of order 6 generated by the Möbius transformations sending $x$ to $y$ and $z$. The other three transformations send $x$ to $y$, to $d(z,x) ≤ h$ and to $y$.

Now let $d(0,y) ≤ h$ be points on the unit circle or real axis in that order. Then the geodesics $d(z,0) ≤ d(z,y) + d(y,0) ≤ 2h$ and $d(a,f(a)) ≤ 2h$ do not intersect and the distance between these geodesics is well defined: there is a unique geodesic line cutting these two geodesics orthogonally and the distance is given by the length of the geodesic segment between them. It is evidently invariant under real Möbius transformations. To compare the cross ratio and the distance between geodesics, Möbius invariance allows the calculation to be reduced to a symmetric configuration. For $f$, take $z, w$. Then $[z,w]$ where $a, b, c, d$. On the other hand, the geodesics $g$ and $(g(a),g(b);g(c),g(d)) = (a,b:c,d)$ are the semicircles in the upper half plane of radius $z$ and $C\\{0,1\}$. The geodesic which cuts them orthogonally is the positive imaginary axis, so the distance between them is the hyperbolic distance between $g(a) = 0, g(b) = 1, g(c) = λ, g(d) = ∞$ and $g$, $a$. Let $b$, then $c$, so that there is a constant $d$ such that, if $a$, then

$$d([a,d];[b,c]) - C \le \log (a,b;c,d) \le d([a,d];[b,c]) + C,$$

since $b$ is bounded above and below in $c$. Note that $d$ are in order around the unit circle if and only if $d$.

A more general and precise geometric interpretation of the cross ratio can be given using projections of ideal points on to a geodesic line; it does not depend on the order of the points on the circle and therefore whether or not geodesic lines intersect.
 * If $a$ and $b$ are the feet of the perpendiculars from $c$ and $λ$ to the geodesic line $1 – λ$, then $λ^{−1}$.

Since both sides are invariant under Möbius transformations, it suffices to check this in the case that $λ$, $1 – λ^{−1}$, $λ(λ – 1)^{−1}$ and $(1 – λ)^{−1}$. In this case the geodesic line is the positive imaginary axis, right hand side equals $S_{4}$, $S_{3}$ and $(a,b)(c,d), (a,c)(b,d)$. So the left hand side equals $(a,d)(b,c)$. Note that $S_{4}$ and $S_{3}$ are also the points where the incircles of the ideal triangles $a, b, c, d$ and $[a,b]$ touch $[c,d]$.

Proof of theorem
A homeomorphism $0 < r < R$ of the circle is quasisymmetric if there are constants $a = –R, b = −r, c = r, d = R$ such that

$$\displaystyle{{|F(z_1)-F(z_2)|\over |F(z_1)-F(z_3)|} \le a {|z_1-z_2|^b\over |z_1-z_3|^b}.}$$

It is quasi-Möbius is there are constants $λ = (a,b;c,d) = (R + r)^{2}/4rR = (t + 1)^{2}/4t$ such that

$$\displaystyle{|(F(z_1),F(z_2);F(z_3),F(z_4))| \le c |(z_1,z_2;z_3,z_4)|^d,}$$

where

$$\displaystyle{ (z_1,z_2;z_3,z_4)={(z_1-z_3)(z_2-z_4)\over(z_2-z_3)(z_1-z_4)}}$$

denotes the cross-ratio.

It is immediate that quasisymmetric and quasi-Möbius homeomorphisms are closed under the operations of inversion and composition.

If $t = R/r > 1$ is quasisymmetric then it is also quasi-Möbius, with $[a,d]$ and $[b,c]$: this follows by multiplying the first inequality for $r$ and $R$. Conversely any quasi-Möbius homeomorphism $ir$ is quasisymmetric. To see this, it can be first be checked that $iR$ (and hence $d(ir,iR) = log R/r = log t$) is Hölder continuous. Let $s = log t$ be the set of cube roots of unity, so that if $λ = cosh^{2}(s/2)$ in $C > 0$, then $(a,b;c,d) > 1$. To prove a Hölder estimate, it can be assumed that $log[cosh(x)/expx)] = log (1 + exp(–2x))/2$ is uniformly small. Then both $x ≥ 0$ and $a, b, c, d$ are greater than a fixed distance away from $(a,b;c,d) > 1$ in $p$ with $q$, so the estimate follows by applying the quasi-Möbius inequality to $c$. To verify that $d$ is quasisymmetric, it suffices to find a uniform upper bound for $ab$ in the case of a triple with $d(p,q) = | log | (a,b;c,d) ||$, uniformly small. In this case there is a point $a = 0$ at a distance greater than 1 from $b = ∞$, $c = x$ and $d = 1$. Applying the quasi-Möbius inequality to $| log | x ||$, $p = | x | i$, $q = i$ and $| log | x ||$ yields the required upper bound. To summarise:

 * A homeomorphism of the circle is quasi-Möbius if and only if it is quasisymmetric. In this case it and its inverse are Hölder continuous. The quasi-Möbius homeomorphisms form a group under composition.

To prove the theorem it suffices to prove that if $p$ then there are constants $q$ such that for $abc$ distinct points on the unit circle

$$\displaystyle{|(F(a),F(b);F(c),F(d))| \le A |(a,b;c,d)|^B.}$$

It has already been checked that $abd$ (and is inverse) are continuous. Composing $ab$, and hence $F$, with complex conjugation if necessary, it can further be assumed that $a, b > 0$ preserves the orientation of the circle. In this case, if $c, d > 0$ are in order on the circle, so too are there images under $F$; hence both $c = a^{2}$ and $d = b$ are real and greater than one. In this case

$$(F(a),F(b);F(c),F(d)) \le A (a,b;c,d)^B.$$

To prove this, it suffices to show that $(z_{1},z_{3},z_{4})$. From the previous section it suffices show $(z_{2},z_{4},z_{3})$. This follows from the fact that the images under $F$ of $F$ and $F^{−1}$ lie within $S$-neighbourhoods of $a ≠ b$ and $S$; the minimal distance can be estimated using the quasi-isometry constants for $| a − b | = 2 sin \pi/3 = \sqrt{3}$ applied to the points on $x – y$ and $x$ realising $y$.

Adjusting $a, b$ and $S$ if necessary, the inequality above applies also to $a ≠ b$. Replacing $x, a, y, b$, $F$, $| F(x) − F(y) | / | F(x) − F(z) |$ and $| x − z | = | x − y |$ by their images under $w$, it follows that

$$A^{-1} |(a,b;c,d)|^{-B} \le |(F(a),F(b);F(c),F(d))| \le A |(a,b;c,d)|^B$$

if $x$, $y$, $z$ and $x$ are in order on the unit circle. Hence the same inequalities are valid for the three cyclic of the quadruple $w$. If $y$ and $z$ are switched then the cross ratios are sent to their inverses, so lie between 0 and 1; similarly if $F = ∂f$ and $A, B > 0$ are switched. If both pairs are switched, the cross ratio remains unaltered. Hence the inequalities are also valid in this case. Finally if $a, b, c, d$ and $F$ are interchanged, the cross ratio changes from $f$ to $F$, which lies between 0 and 1. Hence again the same inequalities are valid. It is easy to check that using these transformations the inequalities are valid for all possible permutations of $F$, $a,b, c,d$, $F$ and $(a,b;c,d)$, so that $(F(a),F(b);F(c),F(d))$ and its inverse are quasi-Möbius homeomorphisms.

Busemann functions and visual metrics for CAT(-1) spaces
Busemann functions can be used to determine special visual metrics on the class of CAT(-1) spaces. These are complete geodesic metric spaces in which the distances between points on the boundary of a geodesic triangle are less than or equal to the comparison triangle in the hyperbolic upper half plane or equivalently the unit disk with the Poincaré metric. In the case of the unit disk the chordal metric can be recovered directly using Busemann functions $log (F(a),F(b);F(c),F(d)) ≤ B log (a,b;c,d) + C$ and the special theory for the disk generalises completely to any proper CAT(-1) space $d([F(a),F(b)],[F(c),F(d)]) ≤ P d([a,b],[c,d]) + Q$. The hyperbolic upper half plane is a CAT(0) space, as lengths in a hyperbolic geodesic triangle are less than lengths in the Euclidean comparison triangle: in particular a CAT(-1) space is a CAT(0) space, so the theory of Busemann functions and the Gromov boundary applies. From the theory of the hyperbolic disk, it follows in particular that every geodesic ray in a CAT(-1) space extends to a geodesic line and given two points of the boundary there is a unique geodesic $f$ such that has these points as the limits $[a,b]$. The theory applies equally well to any CAT($[c,d]$) space with $h$ since these arise by scaling the metric on a CAT(-1) space by $[F(a),F(b)]$. On the hyperbolic unit disk $[F(c),F(d)]$ quasi-isometries of $f$ induce quasi-Möbius homeomorphisms of the boundary in a functorial way. There is a more general theory of Gromov hyperbolic spaces, a similar statement holds, but with less precise control on the homeomorphisms of the boundary.

Applications in percolation theory
More recently Busemann functions have been used by probabilists to study asymptotic properties in models of first-passage percolation and directed last-passage percolation.

title: Busemann function


text:
City Link (company)

City Link was a British next day courier company, operating from 1969 to 2015. It was based in Coventry, West Midlands with offices in other parts of the country. On 24 December 2014, the company entered administration. Ernst & Young (EY) was appointed as the administrator, and immediately ceased accepting parcels from customers. City Link officially ceased trading on 1 January 2015.

History
City Link was founded in 1969 by two minicab drivers, Harry Hammond-Seaman and Jo Burt, as a subsidiary of Orbit Cargo Services Ltd, to provide a cross-town transfer service for British Rail's Red Star Parcels. At the time, Red Star only operated on direct passenger trains, so a sender in Brighton could not dispatch a parcel to Norwich, as a transfer had to be undertaken in London between Victoria Station and Liverpool Street station.

In addition, the Red Star service was limited to station-to-station only. Following increasing demand from its customers, City Link established a nationwide network of delivery agents to collect parcels from the local station and deliver them to the consignee.

With one depot and three employees, the company had a growing customer base, turning over £100 thousand per year. Yet by 1971, profits were derisory, and resultant cash-flow difficulties meant that Orbit Cargo decided to sell the business. The new owner, Bob Thomas, snapped up 75% of the company for £75 and became managing director, a position he occupied for the next 20 years. To gain greater market share, Thomas successfully introduced the franchise concept (the first to do so in the UK parcels sector) by encouraging delivery agents to become franchisees and solicit new business from companies in their areas.

A significant increase in sales led to further cash-flow difficulties, and by June 1989, Thomas had agreed to sell the company to Securiguard Group plc. Being a member of a publicly limited company posed several cultural changes, yet the new owners largely left City Link to generate profits. By 1990, sales had reached £32 million, with a network of 36 depots and over 600 employees.

In July 1993, it was announced that Rentokil plc had successfully made a hostile bid for Securiguard for £76 million, now City Link’s fourth owner since its conception. Unlike Securiguard, Rentokil was uncomfortable with owning a network primarily consisting of franchisees. Whilst they benefitted from profits emanating from City Link, they also coveted the income that the franchisees were generating and were determined to find methods of attaining it for themselves. Keen to attain greater market share and hoping to emulate DHL (part of Deutsche Post AG), Rentokil directors went on the acquisition trail.

On 20 November 2005, Rentokil Initial announced that as a part of its expansion programme, it would buy back the City Link franchises. This was completed in March 2007. In addition, Target Express was identified as an appropriate addition to merge with City Link. The acquisition, coupled with the franchisee by-back was to become a catastrophe for both businesses.

Acquisition of Target Express
Exactly one year after Rentokil Initial announced its City Link franchise buyback scheme, it finished negotiations with the shareholders of Target Express to purchase the company on 20 November 2006. Target was officially acquired, and taken over by City Link on 1 March 2007.

As a part of its merger with Target Express, City Link was planning to close 42 of the 110 combined branches in the first half of 2008, to provide a single integrated network. However, at the end of February 2008, managing director Petar Cvetkovic announced a suspension of further integration of Target Express depots until January 2009.

The merger of the two units proved problematic, particularly due to depots running different computer systems, damaging profits at parent company Rentokil Initial. Rentokil Initial turned down several offers for the loss making unit in May 2008, instead intending to turn around City Link's fortunes.

By November 2011, City Link was still losing money, annual figures released in July show losses growing to £17.8 million, and revenues down 13.5%, and a November 2011 trading statement stated losses of £25 million in the first nine months. Rentokil Initial sold the business to corporate restructuring firm Better Capital in April 2013.

Administration
Late on 24 December 2014, City Link went into administration, and stopped accepting parcels, whilst its administrators Ernst & Young carried out an assessment of the company.

On 25 December 2014, the RMT union was reported to have demanded "urgent talks" with the government and administrators about the collapse of the company. The government said it was unable to intervene in the administration process, although the business secretary said he would meet the union. RMT demanded government talks over the collapse.

On 31 December, the company announced that 2,356 jobs would be lost, leaving only 370 staff employed. The collapse of City Link also left 1,500 self employed drivers out of pocket, with no recourse or protection. A petition was created by the sub contractors to try to get the collapse of City Link debated in the House of Commons.

Rescue offer rejected
BBC News reported on 31 December 2014 that the administrators had rejected an offer from an unnamed consortium, claiming that the possible buyer "offered no money up front and significantly undervalued the assets to be acquired". After confirmation that the rescue had failed, Mick Cash, general secretary of the RMT union, said: "Pulling the plug on any efforts to save City Link is a disgraceful and cynical betrayal that will wreck the lives of our members, many of whom are owed thousands of pounds".

Reports
The Telegraph website reported on 21 March 2015 that "City Link’s creditors are likely to see less than 2p in the pound on any money they are owed". MPs were expected to publish a report on the impact of City Link's closure on employment on 23 March 2015.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: City Link (company)


text:
Turn Against This Land

Turn Against This Land is the debut album from the London-based band Dogs, released in 2005.

Critical reception
The Guardian wrote that "at times - especially on 'Tarred and Feathered' - the sentiments being expressed somehow manage to sound more vital than the roughly hewn punk-pop used to express them." PopMatters wrote that "there is nothing new or surprising about Dogs." NME wrote that "if it’s bug-eyed reprobates with angry hearts powering aneurysm-inducing buzzsaw pop [that you are seeking], then you’ve just found your summer’s soundtrack." The Encyclopedia of Popular Music praised the album, writing that it lived up to the hype generated by the band's singles and live show.

Track listing

 * 1) "London Bridge"  (3:41)
 * 2) "Selfish Ways"  (3:52)
 * 3) "Donkey"  (3:32)
 * 4) "End of an Era"  (3:48)
 * 5) "She's Got a Reason"  (3:26)
 * 6) "It's Not Right"  (3:08)
 * 7) "Tuned to a Different Station"  (4:07)
 * 8) "Tarred and Feathered"   (3:51)
 * 9) "Wait"   (3:53)
 * 10) "Heading for an Early Grave"  (3:07)
 * 11) "Red"   (5:22)
 * 12) "Turn Against This Land"  (1:46)

title: Turn Against This Land


text:
Talk:Free style

'''This guy so called 344 can't spell for s#$t!!!!!!

"I still don't know what free style is??? The definition doesn't make sense!

title: Talk:Free style


text:
Wikipedia:Articles for deletion/Skycutting

 * The following discussion is an archived debate of the proposed deletion of the article below. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in a deletion review).  No further edits should be made to this page.

The result of the debate was delete. -R. fiend 17:41, 12 October 2005 (UTC)

Skycutting
Neologism, non-notable. Also see Articles for deletion/Wittbeat. C HAIRBOY (☎) 00:31, 7 October 2005 (UTC)
 * Delete per nomination. C HAIRBOY (☎) 00:31, 7 October 2005 (UTC)
 * Delete -- ( ☺ drini ♫ | ☎ ) 03:44, 7 October 2005 (UTC)
 * Delete as nn neologism. MCB 02:19, 8 October 2005 (UTC)

 * The above discussion is preserved as an archive of the debate. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in a deletion review). No further edits should be made to this page.


text:
Conjugate points

In differential geometry, conjugate points or focal points are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole are connected by any meridian. Another viewpoint is that conjugate points tell when the geodesics fail to be length-minimizing. All geodesics are locally length-minimizing, but not globally. For example on a sphere, any geodesic passing through the north-pole can be extended to reach the south-pole, and hence any geodesic segment connecting the poles is not (uniquely) globally length minimizing. This tells us that any pair of antipodal points on the standard 2-sphere are conjugate points.

Definition
Suppose p and q are points on a pseudo-Riemannian manifold, and $$\gamma$$ is a geodesic that connects p and q. Then p and q are conjugate points along $$\gamma$$ if there exists a non-zero Jacobi field along $$\gamma$$ that vanishes at p and q.

Recall that any Jacobi field can be written as the derivative of a geodesic variation (see the article on Jacobi fields). Therefore, if p and q are conjugate along $$\gamma$$, one can construct a family of geodesics that start at p and almost end at q. In particular, if $$\gamma_s(t)$$ is the family of geodesics whose derivative in s at $$s=0$$ generates the Jacobi field J, then the end point of the variation, namely $$\gamma_s(1)$$, is the point q only up to first order in s. Therefore, if two points are conjugate, it is not necessary that there exist two distinct geodesics joining them.

For Riemannian geometries, beyond a conjugate point, the geodesic is no longer locally the shortest path between points, as there are nearby paths that are shorter. This is analogous to the Earth's surface, where the geodesic between two points along a great circle is the shortest route only up to the antipodal point; beyond that, there are shorter paths.

Beyond a conjugate point, a geodesic in Lorentzian geometry may not be maximizing proper time (for timelike geodesics), and the geodesic may enter a region where it is no longer unique or well-defined. For null geodesics, points beyond the conjugate point are now timelike separated.

Up to the first conjugate point, a geodesic between two points is unique. Beyond this, there can be multiple geodesics connecting two points.

Suppose we have a Lorentzian manifold with a geodesic congruence. Then, at a conjugate point, the expansion parameter θ in Raychaudhuri's equation becomes negative infinite in a finite amount of proper time, indicating that the geodesics are focusing to a point. This is because the cross-sectional area of the congruence becomes zero, and hence the rate of change of this area (which is what θ represents) diverges negatively.

Examples

 * On the sphere $S^2$, antipodal points are conjugate.
 * On the real coordinate space $$\mathbb{R}^n$$, there are no conjugate points.
 * On Riemannian manifolds with non-positive sectional curvature, there are no conjugate points.

title: Conjugate points


text:
John W. Maher

John W. Maher (June 10, 1866 – May 10, 1917) was an American politician and a veterinarian. He served as a member of the South Dakota House of Representatives.

Biography
John W. Maher was born near Freeport, Illinois to Irish immigrants, James Thomas Maher (January 1, 1842 – March 10, 1923) and Anna (Farley) Maher (November 11, 1839 – September 3, 1907). In 1873, Maher came with his family from Illinois to Dakota Territory, settling in Union County. On May 31, 1887, he married Barbara A. Cooper (August 6, 1861 – August 29, 1937), the daughter of Prussian immigrants. Two of the couple's five children died in infancy.

Maher engaged in business in Beresford, South Dakota and learned that he did not like it. In spite of a lack of college training, Maher obtained a veterinary license; and soon, Maher was recognized as a good practitioner of veterinary medicine. Maher built a hospital near his home and established a widely used line of veterinary medicines.

In the fall of 1911, Maher was one of sixty or seventy men from Beresford who attended a meeting in Sioux Falls, South Dakota. The meeting was called by Dr. Hanson of Brookings, South Dakota for the purpose of organizing plans for a new highway, which would pass from Winnipeg, Manitoba, Canada to the Gulf of Mexico. The highway was eventually known as US Highway Number 77, which was completed in the fall of 1939.

In 1912, Maher was elected to represent the first district of the state Legislature in the South Dakota House of Representatives. He spent his last three years in failing health and used a wheelchair during that time. Maher's health problems were attributed to a strenuous life. He died peacefully in his sleep. During the hour of John W. Maher's funeral, all businesses in Beresford, South Dakota were closed by proclamation of the Mayor.

title: John W. Maher


text:
Barely Legal (album)

Barely Legal is the debut studio album by Swedish rock band The Hives, released on 22 September 1997 through Burning Heart Records. The album's second track, "A.K.A. I-D-I-O-T", had a video produced, but it was not commercially released until it was included as an extra on some pressings of Your New Favourite Band. The band later recorded an EP by the same name. The album sleeve features three quotations from fake newspaper reviews. These quotations are originally from British newspapers referring to the Aldous Huxley novel Brave New World.

Personnel

 * Howlin' Pelle Almqvist – vocals
 * Nicholaus Arson – lead guitar/backing vocals
 * Vigilante Carlstroem – rhythm guitar/backing vocals
 * Dr. Matt Destruction – bass guitar
 * Chris Dangerous – drums

title: Barely Legal (album)


text:
Wikipedia:Articles for deletion/Dark Tide: Siege

 * The following discussion is an archived debate of the proposed deletion of the article below. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or on a Votes for Undeletion nomination).  No further edits should be made to this page.

The result of the debate was NO CONSENSUS. -Doc (?) 16:25, 14 October 2005 (UTC)

Dark Tide: Siege
Novel was actually never completed. It "was to have been" part of a trilogy; but it "isn't"; therefore it doesn't exist. Perhaps the info can be added to another article and this turned into a redirect. Your thoughts are appreciated. paul klenk talk 00:33, 7 October 2005 (UTC)

 * Delete - A non-existent novel doesn't merit an article. I agree with adding content to another article. Sunray 02:17, 7 October 2005 (UTC)
 * Delete, nonexistent. Gazpacho 02:20, 7 October 2005 (UTC)
 * Delete as per nom. --Dvyost 02:46, 7 October 2005 (UTC)
 * Delete per nom. Would be different if it were the lost novel of a great writer. -- SCZenz 02:53, 7 October 2005 (UTC)
 * Merge with New Jedi Order, possibly under a heading of "Cancelled Novels". The Knightfall Trilogy could also be merged into this section. Saberwyn 03:22, 7 October 2005 (UTC)
 * Keep, of interest to users. Kappa 06:44, 7 October 2005 (UTC)
 * Delete. Not just StarWarscruft, non-existent StarWarscruft. --Calton | Talk 07:33, 7 October 2005 (UTC)
 * lol. Delete per above. Dottore So 10:51, 7 October 2005 (UTC)
 * Keep, factual and informative. Hipocrite - « Talk » 13:36, 7 October 2005 (UTC)
 * Merge to the series it was to have been art of. Rich Farmbrough 15:03, 7 October 2005 (UTC)
 * Merge and redirect to the series it would have been part of. --Cel e stianpower hablamé 15:51, 7 October 2005 (UTC)
 * Delete. I think we should raise the inclusion bar very high for things that don't actually exist. Gamaliel 18:22, 7 October 2005 (UTC)
 * Delete per Gama. Very high indeed. Marskell 20:41, 7 October 2005 (UTC)
 * Delete per nominator and Gamaliel. Quale 22:39, 7 October 2005 (UTC)
 * Keep I disagree with the gut instinct to label unpublished novels non-encyclopedic. Just because something gets cancelled doesn't mean that people were unaware of it or uninterested in it. Now, in general, I question the need for every single New Jedi Order novel to have its own wikipedia article; but if that's the basis for deletion, then I argue for a Merge of all these novel stubs into the primary article. Justin Bacon 00:34, 8 October 2005 (UTC)
 * Merge to the series article. Amren (talk) 13:50, 8 October 2005 (UTC)
 * The above discussion is preserved as an archive of the debate. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in an undeletion request). No further edits should be made to this page.


text:
Hadamard space

In geometry, an Hadamard space, named after Jacques Hadamard, is a non-linear generalization of a Hilbert space. In the literature they are also equivalently defined as complete CAT(0) spaces.

A Hadamard space is defined to be a nonempty complete metric space such that, given any points $$x$$ and $$y,$$ there exists a point $$m$$ such that for every point $$z,$$ $$d(z, m)^2 + {d(x, y)^2 \over 4} \leq {d(z, x)^2 + d(z, y)^2 \over 2}.$$

The point $$m$$ is then the midpoint of $$x$$ and $$y:$$ $$d(x, m) = d(y, m) = d(x, y)/2.$$

In a Hilbert space, the above inequality is equality (with $$m = (x+y)/2$$), and in general an Hadamard space is said to be if the above inequality is equality. A flat Hadamard space is isomorphic to a closed convex subset of a Hilbert space. In particular, a normed space is an Hadamard space if and only if it is a Hilbert space.

The geometry of Hadamard spaces resembles that of Hilbert spaces, making it a natural setting for the study of rigidity theorems. In a Hadamard space, any two points can be joined by a unique geodesic between them; in particular, it is contractible. Quite generally, if $$B$$ is a bounded subset of a metric space, then the center of the closed ball of the minimum radius containing it is called the circumcenter of $$B.$$ Every bounded subset of a Hadamard space is contained in the smallest closed ball (which is the same as the closure of its convex hull). If $$\Gamma$$ is the group of isometries of a Hadamard space leaving invariant $$B,$$ then $$\Gamma$$ fixes the circumcenter of $$B$$ (Bruhat–Tits fixed point theorem).

The basic result for a non-positively curved manifold is the Cartan–Hadamard theorem. The analog holds for a Hadamard space: a complete, connected metric space which is locally isometric to a Hadamard space has an Hadamard space as its universal cover. Its variant applies for non-positively curved orbifolds. (cf. Lurie.)

Examples of Hadamard spaces are Hilbert spaces, the Poincaré disc, complete real trees (for example, complete Bruhat–Tits building), $(p, q)$-space with $$p, q \geq 3$$ and $$2 p q \geq p + q,$$ and Hadamard manifolds, that is, complete simply-connected Riemannian manifolds of nonpositive sectional curvature. Important examples of Hadamard manifolds are simply connected nonpositively curved symmetric spaces.

Applications of Hadamard spaces are not restricted to geometry. In 1998, Dmitri Burago and Serge Ferleger used CAT(0) geometry to solve a problem in dynamical billiards: in a gas of hard balls, is there a uniform bound on the number of collisions? The solution begins by constructing a configuration space for the dynamical system, obtained by joining together copies of corresponding billiard table, which turns out to be an Hadamard space.

title: Hadamard space


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Riverview (Gatineau)

Riverview is an historic manor at 432, boulevard Alexandre-Taché in the Hull sector of Gatineau, Quebec, Canada. It was originally built by the lumber baron David Moore in 1865. The sprawling Jacobean structure is a noted landmark. The house later became home to Edward S. Skead and his family, who were also in the lumber business.

The city slowly encroached on the manor and it lost its exclusive status. After the First World War, it was sold and became a popular nightclub known as the Homestead Inn. In 1939, it changed roles again becoming the Ville-Joie-Ste-Thérèse orphanage. As orphanages were phased out in the 1960s, the facility became a nursery, but this closed in 1972. Today the building is the home of the Gatineau campus of the Conservatoire de musique et d'art dramatique du Québec.

title: Riverview (Gatineau)


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Dowel

A dowel is a cylindrical shape made of wood, plastic, or metal. In its original manufactured form, a dowel is long and called a dowel rod, which are often cut into shorter dowel pins. Dowels are commonly used as structural reinforcements in cabinet making and in numerous other applications, including:
 * Furniture shelf supports
 * Moveable game pieces (i.e. pegs)
 * Hangers for items such as clothing, key rings, and tools
 * Wheel axles in toys
 * Detents in gymnastics grips
 * Supports for tiered wedding cakes

Manufacturing process
The traditional tool for making dowels is a dowel plate, an iron (or better, hardened tool steel) plate with a hole having the size of the desired dowel. To make a dowel, a piece of wood is split or whittled to a size slightly bigger than desired and then driven through the hole in the dowel plate. The sharp edges of the hole shear off the excess wood.

A second approach to cutting dowels is to rotate a piece of oversized stock past a fixed knife, or alternatively, to rotate the knife around the stock. Machines based on this principle emerged in the 19th century. Frequently, these are small bench-mounted tools.

For modest manufacturing volumes, wood dowels are typically manufactured on industrial dowel machines based on the same principles as the rotary cutters described above. Such machines may employ interchangeable cutting heads of varying diameters, thus enabling the machines to be quickly changed to manufacture different dowel diameters. Typically, the mechanism is open-ended, with material guides at the machine's entry and exit to enable fabrication of continuous dowel rods of unlimited length. Since the 19th century, some of these dowel machines have had power feed mechanisms to move the stock past the cutting mechanism.

Application
When dowels are glued into blind holes, a very common case in dowel-based joinery, there must be a path for air and excess glue to escape when the dowel is pressed into place. If no provision is made to relieve the hydraulic pressure of air and glue, hammering the dowel home or clamping the joint can split the wood. An old solution to this problem is to plane a flat on the side of the dowel; some sources suggest planing the flat on the rough stock before the final shaping of the round dowel. Some dowel plates solve the problem by cutting a groove in the side of the dowel as it is forced through; this is done by a groove screw, a pointed screw intruding from the side into the dowel cutting opening.

When two pieces of wood are to be joined by dowels embedded in blind holes, there are numerous methods for aligning the holes. For example, pieces of shot may be placed between the wood pieces to produce indentations when the pieces are clamped together; after the clamp is released, the indentations indicate the center points for drilling.

History
The word dowel was used in Middle English; it appears in Wycliffe's Bible translation (circa 1382–1395) in a list of the parts of a wheel: "...and the spokis, and dowlis of tho wheelis..."

title: Dowel


text: User talk:69.0.56.169

title: User talk:69.0.56.169


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Manitoba Liberal Party candidates in the 1999 Manitoba provincial election

One member of the Manitoba Liberal Party was elected to the Legislative Assembly of Manitoba in the 1999 provincial election. Some of the party's candidates have their own biography pages; information about others may be found here.

Bob Brigden (Arthur-Virden)
Brigden is a farmer near Melita, Manitoba. He was elected as a councillor in the Rural Municipality of Brenda in 1998, and was the first declared candidate for the 1999 Progressive Conservative nomination in Arthur-Virden. He later withdrew from that contest, and sought and won the Liberal Party nomination instead. Newspaper accounts do not indicate why he changed parties in the buildup to the campaign. Brigden did not stand for re-election to council in 2002.

J. Deborah Shiloff (Assiniboia)
Shiloff is a neuroscientist, a Scientific Support Officer at the National Research Council of Canada's Winnipeg Institute for Biodiagnostics, and a member of the Neuroethics New Emerging Team (NET). She received 1,136 votes (11.56%) in the 1999 election, finishing third against New Democratic Party candidate Jim Rondeau.

Peter Logan (Brandon East)
Logan has campaigned for Brandon City Council on two occasions. He sold computerized cash register systems in 1999, and could not campaign full-time until the last two weeks of the election. He later became president of the federal Liberal Party riding association in Brandon—Souris, and was one of Paul Martin's nominators for the party leadership in 2003. Logan supported Michael Ignatieff's leadership bid in 2006.

Lisa Roy (Brandon West)
Roy (now Lisa Blixhavn) is a teacher, and an active curler. She led the 1998 Manitoba women's junior champions at the 1998 Canadian Junior Curling Championships in Calgary, Alberta, and was a member of the runner-up team in the 2005 women's provincial curling championship. She has remained active in the sport after suffering a serious eye injury in a 2005 accident.

Roy was a university student during the 1999 campaign. She received 407 votes (4.09%), finishing third against New Democratic Party candidate Scott Smith.

Chris Hlady (Concordia)
Hlady has a Bachelor of Fine Arts from the University of Manitoba. He has been a comic book illustrator, and has done video work. He received 444 votes (5.47%) in 1999, finishing third against New Democratic Party leader Gary Doer.

Malli Aulakh (Fort Whyte)
Aulakh was one of five candidates for the federal Liberal Party nomination for Winnipeg South in 1993. He was eliminated after the first ballot and gave his support to rival candidate Reg Alcock, the eventual winner (WFP, 25 March 1993). He later attempted to challenge Kevin Lamoureux for the Winnipeg Centre Liberal nomination in 2000, but was unable to do so when the party ruled he had missed a filing deadline (WFP, 30 September 2000).

He received 1,202 votes (11.45%) in the 1999 election, finishing third against Progressive Conservative John Loewen. In 2003, Aulakh was named to the board of directors of the Winnipeg Public Library (WFP, 24 March 2003).

Vic Wieler (Kirkfield Park)
Wieler campaigned ran for the Manitoba legislature in the 1995 provincial election, and finished second against Progressive Conservative incumbent Eric Stefanson. He campaigned against Stefanson again in the 1999 election, and finished third.

In 2000, he was nominated as that Liberal candidate for a by-election in Kirkfield Park against new Progressive Conservative Party Stuart Murray. He finished second with 2,158 votes (26.64%).

Previously a high-school principal, Wieler had become a financial planner by the time of the 2000 by-election. He was also president of the Rotary-Club of Winnipeg Assiniboine at the time of the by-election (Winnipeg Free Press, 11 May 2000). In 2003, he became a director of the Mennonite Central Committee's Furniture Thrift Store (Winnipeg Free Press, 12 November 2003).

Ajay Chopra (Point Douglas)
Chopra was awarded the National Indo-Canadian Council Distinguished Youth Entrepreneurship Award in 1998. He was 22 years old at the time of the 1999 election, and finished a credible second against New Democratic Party incumbent George Hickes with 1,336 votes (21.35%).

Chopra was a special assistant to the federal Minister of Justice and Attorney General in 2002, and worked on Phil Fontaine's campaign to lead the Assembly of First Nations (AFN) in 2003. Although he is not himself aboriginal, Chopra currently serves as the AFN's intergovernmental affairs advisor.

Betty Ann Watts (Radisson)
Watts was candidate for the Manitoba Liberal Party in two elections, 1999 in Radisson and 2003 in Transcona. Watts was a trustee in the Transcona-Springfield School Division from 1989 to 1998, and served as its chair prior to the 1998 municipal election (in which she was not a candidate). She supported greater parental involvement in school activities, and played an organizational role in Ukrainian language bilingualism programs. In 1996, she was elected chair of the Manitoba Multicultural Resources Centre. She has also worked as Managing Editor for The Cottager magazine, and coordinated the 2007 Magazines Mean Business conference.

Cecilia Connelly (Rossmere)
Connelly has been a frequent candidate for the Liberal Party in the Rossmere constituency. Her poor showing in 1999 may be explained by the fact that the New Democratic and Progressive Conservative parties were engaged in a tightly-fought contest, with both parties siphoning Liberal support.

Vibart C. Stewart (Transcona)
Stewart and his wife helped supply replacement workers to the Tache Nursing Centre in 1996, at a time with the centre's employees were on strike. Vibart's wife, ironically, was herself one of the strikers. He received 713 votes (8.10%) in 1999, finishing third against New Democratic Party incumbent Daryl Reid.

Rochelle Zimberg (Tuxedo)
Rochelle Zimberg has a master's degree in international relations theoryt. She was executive director of the Manitoba Association of Urban Municipalities during the 1990s, and argued in 1993 that the provincial Municipal Act was out of date and in need of revision. She later sought a balanced approach on the use of Video Lottery Terminal revenues by municipal governments. In 1997, she was appointed to an advisory committee overseeing the distribution of the Canadian Red Cross, Manitoba Flood Appeal Fund. She was inducted into the Federation of Canadian Municipalities Roll of Honour in 1999. Zimberg placed third against Manitoba Premier Gary Filmon in 1999'. after the original Liberal candidate stood from the contest. ms Zimberg ran again in 2000 and came in second to the conservative candidate.