Siberian Mathematical Journal, Vol. 47, No. 1, pp. 1–17, 2006 c 2006 Andreev P. D. Original Russian Text Copyright
A. D. ALEXANDROV’S PROBLEM FOR CAT(0)-SPACES P. D. Andreev
UDC 514.763.254
Abstract: We solve the well-known problem by A. D. Alexandrov for nonpositively curved spaces. Let X be a geodesically complete locally compact space nonpositively curved in the sense of Alexandrov and connected at infinity. The main theorem reads as follows: Each bijection f : X → X such that f and the inverse f −1 of f preserve distance 1 is an isometry of X. Keywords: Alexandrov’s problem, nonpositively curved space, isometry
§ 1. Introduction In [1], V. N. Berestovski˘ı proved the following characterization for isometries of Alexandrov spaces of negative curvature separated from zero: Theorem 1.1 [1, Theorem 1.1]. Let X be a geodesically complete locally compact CAT(K)-space with curvature at most K, where K < 0, in which all spheres are arcwise connected. Then every bijection f of X onto itself such that f and f −1 map any closed ball of some fixed radius r > 0 onto some closed ball of radius r is an isometry. The proof of Theorem 1.1 bases on the following stronger assertion. Theorem 1.2 [1, Theorem 3.1]. Let X be a geodesically complete locally compact CAT(K)-space which is connected at infinity, and let V ⊂ X × X be the diagonal tube corresponding to a number r > 0. Then the metric of X is uniquely determined by V . Here by the diagonal tube V in a metric space X corresponding to r > 0 we mean the set V := {(x, y) ∈ X × X | |xy| ≤ r} ⊂ X × X, where |xy| is the distance between points x, y ∈ X. Arcwise connectedness of spheres in Theorem 1.1 is equivalent to connectedness of the space at infinity. This means that the complement of every metric ball in X is arcwise connected. The following question was formulated in Berestovski˘ı’s article [1]: Do similar assertions hold for the spaces nonpositively curved in the Alexandrov sense (the so-called CAT(0)-spaces)? This article gives a positive answer to this question. Earlier in [2], the author considered this problem for the spaces nonpositively curved in the Alexandrov sense without overlapping minimizers. It was shown in [3] that each Alexandrov space of curvature bounded above with the local prolongation property and nonoverlapping minimizers is a topological manifold with the components of the metric tensor continuous with respect to distance coordinates; therefore, for such a space, connectedness at infinity is equivalent to the condition n > 1 for the topological dimension n = TopDim(X). The problems we address arise in connection with a problem posed by A. D. Alexandrov in the 1960s. The full statement of Alexandrov’s problem is as follows: Describe metric spaces such that each surjection to itself preserving a fixed distance, for example distance 1, is an isometry. One of the first articles on the topic was written by Beckman and Quarles in 1953 [4]. The Beckman– Quarles theorem states that if n ≥ 2 then each mapping of the Euclidean space En into En preserving distance 1 is an isometry. The author was supported by the Russian Foundation for Basic Research (Grant 04–01–00315). Arkhangel sk. Translated from Sibirski˘ı Matematicheski˘ı Zhurnal, Vol. 47, No. 1, pp. 3–24, January–February, 2006. Original article submitted October 28, 2004. Revision submitted March 3, 2005. c 2006 Springer Science+Business Media, Inc. 0037-4466/06/4701–0001
1
A. D. Alexandrov (for example, see [5]) considered a number of similar problems. In particular, he proved some necessary and sufficient conditions for mappings of space forms preserving the congruence of figures of a fixed class to be isometries. Alexandrov’s investigations into Lobachevski˘ı space were continued by A. V. Kuz minykh in [6]. He showed that if a mapping f of the Lobachevski˘ı space Λn , n ≥ 2, into Λn is such that, for positive numbers a and a , |xy| = a implies |f (x)f (y)| = a then f is an isometry and a = a . Recently, interest in Alexandrov’s problem and the Beckman–Quarles theorem has renewed due to the development of metric geometry. Here we should mention Tyszka’s articles (for example, see [7, 8]), where the Beckman–Quarles theorem is generalized to mappings of complex spaces and to mappings into Rn of the space F n , where F is a field of characteristic 0, and also some discrete versions of this theorem are proven. 1.1. The contents and structure of the article. In the sequel we prove Theorem 1.3. Let X be a geodesically complete locally compact CAT(0)-space connected at infinity and let f : X → X be a bijection. If f is such that |f (x)f (y)| = 1 if and only if |xy| = 1 then f is an isometry of X. Following the logic of [1], we will prove a stronger theorem which states that, under the conditions of Theorem 1.3, the metric of a space is uniquely determined by its diagonal tube corresponding to an arbitrary number r > 0. In what follows, we always assume that r = 1 but preserve V. N. Berestovski˘ı’s term “r-sequence” and the index r in the symbols of r-sequences. Moreover, we also involve the boundary ∂V := {(x, y) ∈ X × X | |xy| = 1} ⊂ X × X and the interior Int V := {(x, y) ∈ X × X | |xy| < 1} ⊂ X × X of the diagonal tube V . Theorem 1.4. Let (X, d) be as in Theorem 1.3 and let V be its diagonal tube corresponding to 1. Suppose that a metric d on X is such that the space (X, d ) with diagonal tube V also meets the hypotheses of Theorem 1.3. Then the following are equivalent: (1) V = V , (2) ∂V = ∂V , (3) Int V = Int V , (4) d = d . For proving that Theorem 1.3 is a corollary to Theorem 1.4, it suffices to consider the metric d = f ∗ d on X: f ∗ d(x, y) = d(f (x), f (y)). Under the conditions of Theorem 1.3, we have ∂V = ∂V . We call every metric d that meets the hypotheses of Theorem 1.3 and has V as its diagonal tube a metric realizing V , or, briefly, a V -metric. In particular, the metric d itself is a metric realizing V . Theorem 1.4 states that d is the only metric on X that realizes V . However, we a priori assume existence of a set of such metrics and proceed with proving the theorem by demonstrating consecutively that an arbitrary V -metric coincides with d on a larger and larger subset of X × X. In § 2, we expose some necessary notions and facts of the theory of nonpositively curved spaces and give the main definitions. The proof of Theorem 1.4 for the spaces nonpositively curved in the Alexandrov sense is the contents of § 3 and § 4. Our general approach to the proof differs from the strategy of [1] where the base step is the reconstruction of the diagonal tube 12 V corresponding to 12 from V . This step is applied to the whole space simultaneously and then is repeated infinitely. In our case, the metric is reconstructed on each individual geodesic. Geodesics in nonpositively curved spaces differ by rank. Usually, by the rank of a geodesic we mean the maximal dimension of a Euclidean space isometrically embedded in X and including this geodesic. Here we somewhat deviate from this custom and by higherrank geodesics we mean geodesics that lie on the boundary of a flat strip embedded in X. If a geodesic 2
does not lie on the boundary of a flat strip then we call it a rank-one geodesic. We make another change to the treatment of the notion of the rank of a geodesic. Divide in addition all geodesics into virtually higher-rank geodesics and strictly rank-one geodesics. The exact definitions are given in 2.3. In § 3, we consider the case of a virtually higher-rank geodesic and in § 4, the case of a strictly rankone geodesic. In each of the situations under consideration, the author involves new constructions: the so-called “tapes” for higher-rank geodesics and “scissors” for rank-one geodesics. The author is grateful to V. N. Berestovski˘ı for useful discussions during preparation of the article and expresses a special gratitude to the referee for his attention to the article and a number of valuable suggestions and remarks. § 2. Main Notions 2.1. Necessary information from the theory of nonpositively curved spaces. In this section, we recall basic definitions and facts of the geometry of nonpositively curved Alexandrov spaces. The detailed information on the geometry of Alexandrov spaces of curvature bounded above can be found in [9–11] and some other sources. Let (X, d) be a metric space. In the sequel, we denote the distance d between points x, y ∈ X by |xy| := d(x, y). Denote the ball of radius ρ with center a point x ∈ X by B(x, ρ) and the corresponding boundary sphere, by S(x, ρ). Given A ⊂ X and ε > 0, the set Nε (A) := {x ∈ X | d(x, a) < ε for some a ∈ A} is understood to be the ε-neighborhood of A. For closed sets A, B ⊂ X, the Hausdorff distance between A and B is by definition dH (A, B) := inf{ε | A ⊂ Nε (B), B ⊂ Nε (A)}; in particular, if there is no ε > 0 such that A ⊂ Nε (B) and B ⊂ Nε (A) then dH (A, B) is taken to be ∞. A geodesic in a metric space (X, d) is by definition the image of a locally isometric mapping c : I → X from some interval I ⊂ R into X, i.e., of such a mapping that the equality d(c(s1 ), c(s2 )) = |s1 − s2 | holds in a neighborhood U of every t ∈ I and for all s1 , s2 ∈ U . The mapping c is called the natural parametrization of the geodesic. In the sequel, we always consider naturally parametrized geodesics without distinguishing between the geodesic itself and its natural parametrization. If we take the whole interval I as U , the geodesic c is called a minimizer. If, moreover, I is a segment [a, b] ⊂ R then we say that c(I) is a segment joining c(a) and c(b). If I = R then the geodesic c is called a complete geodesic. The space (X, d) is called a geodesic metric space if there is a segment for every two points. A geodesic space is called geodesically complete if each geodesic therein can be prolonged to a complete geodesic (not necessarily unique). The Hopf–Rinow theorem (see [12, Theorem 2.3]) implies that a geodesically complete locally compact space is finitely compact, or, in other terms, proper, i.e., every bounded closed set therein is compact. A triangle in a geodesic space is by definition the union of three segments ci : [ai , bi ] → X, i = 1, 3, called the sides of the triangle joining pairwise three points xi , i = 1, 3, called the vertices of the triangle. For a triple of points Δ = (x1 , x2 , x3 ) in X a comparison triangle Δ ⊂ E2 on the Euclidean plane 2 ¯1, x ¯2 , x ¯ 3 corresponding to x1 , x2 , x3 , has side lengths dκ (¯ xi , x ¯ j ) = d(xi , xj ), i, j = 1, 3. E with vertices x A comparison triangle is defined up to a Euclidean isometry. If ci are the sides of the triangle with vertices Δ then the sides of the corresponding comparison triangle Δ are denoted by ¯ci . A point m ∈ Δ corresponds to a point m ∈ Δ if, for some i, there exists a number ti ∈ [ai , bi ] satisfying m = ¯ci (ti ) and m = ci (ti ). A triangle whose vertices form a triple Δ is called thin if the inequality ¯ ), (2.1) d(m, n) ≤ dE (m, n where dE is the Euclidean distance, holds for every pair of its points m, n and the pair of corresponding ¯ of the comparison triangle on the model surface Mκ . points m, n 3
A domain U in a geodesic metric space X is called a 0-domain (or a domain R0 ) if every triangle lying entirely in U is thin. A complete geodesic metric space X is called a nonpositively curved Alexandrov space if each of its points has a neighborhood U that is a 0-domain. A CAT(0)-space is a nonpositively curved Alexandrov space that is a 0-domain. By the Alexander–Bishop theorem [13], every CAT(0)-space is simply connected. 2.2. The boundary of a nonpositively curved space. The notion of boundary of a space nonpositively curved in the Alexandrov sense is well known. There are two essentially different approaches to it, which give the same result. Here we need both definitions. In the cases when the distinctions are inessential, we speak about the boundary in the general sense. Definition 2.1. Geodesic rays c, d : [0, +∞) → X in a metric space X are called asymptotic if the Hausdorff distance between them is finite: Hd(c, d) < +∞. The asymptoticity relation is an equivalence on the set of geodesic rays in X. The set of equivalence classes is called the geodesic ideal boundary of X and is denoted by ∂g X. The union X g = X ∪ ∂g X is called the geodesic ideal closure of X. This terminology corresponds to the notions of geodesic boundary and geodesic closure as introduced in [14]. Given a pair of points y, z ∈ X, the expression [yz] denotes the segment of these points if y, z ∈ X or the geodesic ray starting at y in the direction of z if y ∈ X and z ∈ ∂g X (if X is a locally compact CAT(0)-space then such a ray exists and is unique) or an arbitrary complete geodesic with endpoints at y and z if y, z ∈ ∂g X and such a geodesic exists. The closure X g is naturally endowed by the cone topology, the topology of uniform convergence on bounded sets of segments and rays. The sequence {xn }+∞ n=1 ⊂ X g converges to a point x ∈ X g in the cone topology if, for a distinguished point o ∈ X, the sequence of naturally parametrized segments or rays {[oxn ]}+∞ n=1 converges to the natural parametrization of the segment (ray) [ox] uniformly on bounded sets. The cone topology on X g does not depend on the choice of o. The induced topology on the geodesic ideal boundary ∂g X is also called the cone topology. Definition 2.2. Given a locally compact metric space X, its Kuratowski embedding in the space C(X) of continuous functions on X is defined. Namely, if o ∈ X is a distinguished point then a point x is identified with the distance function dx , defined by the equality dx (y) = |xy| − |ox|. C ∗ (X)
= C(X)/{consts} be the quotient space C(X) by the subspace of constants. Then the Let projection C(X) → C ∗ (X) generates an embedding i : X → C ∗ (X) independent of the distinguished point o. The topology on C ∗ (X) is inherited from the compact-open topology on C(X). We identify X with its image i(X). The closure of i(X) ⊂ C ∗ (X) is called the metric closure of X. The metric closure is denoted by X m , the metric boundary is by definition ∂m X = X m \ X. See [15] for the origin of the term “metric closure.” The functions, constituting the metric boundary, are called horofunctions. They are limits in the sense of compact-open topology for distance functions. A special case of horofunctions is given by the Busemann function. For a geodesic ray c : [0, +∞) → X, the Busemann function βc generated by this ray is defined by the equality βc (y) = lim (|yc(0)| − t). t→+∞
If Φ ∈ ∂m X is a horofunction and x ∈ X is an arbitrary point then the horosphere H S (Φ, x) = {y ∈ X | Φ(y) = Φ(x)} and the horoball H B(Φ, x) = {y ∈ X | Φ(y) ≤ Φ(x)}, i.e., the level and sublevel sets of Φ corresponding to x are defined. 4
If X is a locally compact CAT(0)-space then its geodesic and metric closures coincide in the following sense: The identity mapping IdX admits a unique extension to a homeomorphism X m → X g . For simply connected nonpositively curved smooth Riemannian manifolds, coincidence of the two boundaries was demonstrated in [16]; moreover, the method of proof is applicable to Alexandrov spaces practically verbatim (see [16]). In particular, every horofunction is a Busemann function defined from some geodesic ray and Busemann functions defined from asymptotic rays differ only by a constant. Denote the ideal closure of a CAT(0)-space X by X and denote the ideal boundary of X by ∂∞ X. Definition 2.3. Suppose that a point at infinity ξ ∈ ∂X is defined by a ray c : R+ → X. By the horosphere (horoball ) with center ξ we mean the level set H S (ξ, x) = H S (βc , x) (the sublevel set H B(ξ, x) = H B(βc , x)) of the Busemann function βc . 2.3. The rank of a geodesic. Definition 2.4. Complete geodesics c1 and c2 are called parallel if the Hausdorff distance Hd(c1 , c2 ) between them is finite. Rays included in parallel geodesics define exactly two points of the ideal boundary. Parallel geodesics bound a flat strip in a CAT(0)-space, i.e., a strip of the Euclidean plane isometrically embedded in X. A geodesic c is called a rank-one geodesic if it does not lie on the boundary of a flat strip in X. In this case, c does not admit any parallel geodesics different from c. If c is on the boundary of a flat strip then we say that c is a higher-rank geodesic. Definition 2.5. We say that complete geodesics c and d are joined by an asymptotic chain if there exists a finite sequence of geodesics c := c0 , c1 , . . . , cn := d such that ci−1 and ci are asymptotic to each other in some direction for all i = 1, n. A geodesic c virtually has higher rank if there exists a higher-rank geodesic d that can be joined by an asymptotic chain with c. In particular, a higher-rank geodesic is a virtually higher-rank geodesic itself. If a geodesic is not a virtually higher-rank geodesic then we say that c is strictly rank-one. 2.4. The diagonal tube of a nonpositively curved space. From now on, we assume (X, d) a geodesically complete simply connected locally compact space nonpositively curved in the Alexandrov sense and connected at infinity (or a CAT(0)-space); X = X ∪ ∂∞ X is the ideal closure of X. Suppose that X is endowed with two metrics d and d that have a common diagonal tube V and suppose that both metric spaces (X, d) and (X, d ) meet the hypotheses of Theorem 1.3. If n ∈ N then we consider the diagonal tubes nV := {(x, y) ∈ X × X | |xy| ≤ n}, their boundaries ∂(nV ) := {(x, y) ∈ X × X | |xy| = n}, and interiors Int(nV ) := nV \ ∂(nV ). Lemma 2.1. If two metrics d and d on X satisfy the conditions of Theorem 1.3 and have a common diagonal tube V then the sets nV , ∂(nV ), and Int(nV ) for all n ∈ N are the same for both metrics. Proof. We have (x, y) ∈ 2V if and only if there exists a point z ∈ X with (x, z), (z, y) ∈ V . This holds or does not hold for both metrics d and d simultaneously. By induction, (x, y) ∈ nV if and only if there exists a point z ∈ X satisfying (x, z) ∈ (n − 1)V and (z, y) ∈ V . A pair (x, y) belongs to ∂(2V ) if and only if there exists a unique point z such that (x, z), (z, y) ∈ V . In this case, (x, z), (z, y) ∈ ∂V and z is the midpoint of the segment [xy]. Therefore, d and d have a common boundary ∂(2V ). Furthermore, (x, z) ∈ ∂V if and only if there exists a pair (x, y) ∈ ∂(2V ) such that z is the midpoint of [xy]. Hence, the boundaries ∂V with respect to d and d coincide too. By induction, we prove that the boundaries ∂(nV ) coincide. The equality Int(nV ) = nV \ ∂(nV ) holds for Int(nV ) by definition. 5
Thus, every two V -metrics d and d have the same open and closed balls and spheres of integer radii. Lemma 2.2. Under the conditions of Lemma 2.1, if at least one of the sets V , ∂V , or Int V is the same for the metrics d and d then the remaining two sets coincide as well. Proof. The case of coincidence of the tubes V was considered in Lemma 2.1. Prove the lemma in the two remaining cases. Suppose that d and d have the same boundary ∂V of the diagonal tube. Similar to Lemma 2.1, this implies coincidence of the diagonal tubes 2V . In particular, the metrics have a common system of balls B(x, 2n) of even radii and, hence, by hypothesis and Lemma 2.1, all spheres of the form S(x, 1) and S(x, 2n) with x ∈ X coincide. Now, connectedness at infinity implies that (x, y) ∈ Int V if and only if S(x, 1) ∩ S(y, 1) = ∅ and S(x, 2) ∩ S(y, 1) = ∅. Therefore, the metrics have a common interior Int V . Now, V = Int V ∪ ∂V . Suppose finally that d and d have a common interior Int V of V . Then they have the same interiors Int(nV ) of nV for all n ∈ N. By the geodesic completeness of X, the relation (x, y) ∈ ∂V is fulfilled if and only if (x, y) ∈ / Int V and B(y, 1) ⊂ B(x, 2). Indeed, if 1 < d(x, y) < 2 then the segment [xy] is included in a segment [xz] of length d(x, z) = 12 (3 + d(x, y)). The point z meets the containment z ∈ B(y, 1) \ B(x, 2). Thus, Int V uniquely defines ∂V and hence V . We may now assume that all three relations V , ∂V , and Int V are defined. 2.5. r-Sequences. The notion of r-sequence was introduced in [1]. We apply its following paraphrase. Definition 2.6. By an r-sequence we mean an isometric embedding Z → X of the set of integers Z into X. Denote the segment of an r-sequence {xz }z∈Z between xz1 and xz2 by [xz1 , xz1 +1 , . . . , xz2 ]r . This definition agrees completely with V. N. Berestovski˘ı’s definition but is not constructive. The analysis of r-sequences in [16] and Lemmas 2.1 and 2.2 imply that, knowing the diagonal tube V , we can see whether a sequence of points {xz }z∈Z parametrized by Z is an r-sequence. In other words, all metrics realizing the diagonal tube V of the metric have the same set of r-sequences. Each r-sequence lies on a unique geodesic and, conversely, to define a geodesic, it suffices to point out an r-sequence included therein. Thus, given two V -metrics d and d , the r-sequences define a correspondence τ between their geodesics: a geodesic a in the metric d corresponds to a geodesic a in the metric d if they are defined by a common r-sequence. If all geodesics in (X, d) have rank one then this implies that the correspondence τ is one-to-one. The sequel implies that the property of τ to be one-to-one holds also in the general case of a nonpositively curved space meeting the conditions of Theorem 1.3. Definition 2.7. Two r-sequences {xz }z∈Z and {yz }z∈Z are called parallelly equivalent if the Hausdorff distance between them is finite: Hd({xz }, {yz }) < +∞. This condition is equivalent to the following: There exists a natural n such that (xz , yz ) ∈ Vn for all z ∈ Z. Lemma 2.3. The geodesic defined by an r-sequence {xz }z∈Z has higher rank if and only if there is an r-sequence {yz }z∈Z parallelly equivalent to it, different from {xz±1 }z∈Z and such that (x0 , y0 ) ∈ ∂V . Proof. If a geodesic a has higher rank then it suffices to take y0 to be the interior point in the flat strip that includes a with |x0 y0 | = 1. The geodesic passing through y in parallel to a is defined by a desired r-sequence. Conversely, if {yz }z∈Z is an r-sequence of the kind then it defines a geodesic a parallel to a. By the Flat Strip Lemma (see [16] or [11]), two parallel geodesics in a nonpositively curved space bound a flat strip in X. 6
Corollary 2.1. Let a be a rank-one geodesic in a metric d. Then a is a geodesic and has rank one in an arbitrary metric d realizing V . Proof. The curve a defines a family of r-sequences in which no two r-sequences {xz }z∈Z and {yz }z∈Z that fail to coincide pointwise satisfy the condition (x0 , y0 ) ∈ ∂V . All r-sequences under consideration are parallelly equivalent and form an entire class of parallelly equivalent r-sequences. By the lemma, the geodesics that they define cannot be different parallel curves in any V -metric. Consequently, such geodesics coincide pairwise and with the curve a. 2.6. Horospherical transfer of a metric. As follows from [1, Proposition 3.5], the sets of horoballs and horospheres coincide for every two metrics d and d of X that realize V . This yields the following possibility of transferring a metric. Lemma 2.4. Let curves γ and γ1 be geodesics in two metrics d and d realizing V that are asymptotic in the direction of the ideal point ξ = c(+∞) ∈ ∂X. Suppose that d = d along γ. Then again d = d along γ1 . Proof. Consider points x1 , y1 ∈ γ1 and horospheres −1
H S (ξ, x1 ) = βc−1 (x1 ) = β c (x1 ) and
−1
H S (ξ, y1 ) = βc−1 (y1 ) = β c (y1 ),
where βc and βc are Busemann functions corresponding to the ray c in d and d . Put x = γ ∩ H S (ξ, x1 ) and y = γ ∩ H S (ξ, y1 ). Then d (x1 , y1 ) = |βc (x1 ) − βc (y1 )| = |βc (x) − βc (y)| = d (x, y) = d(x, y) = |βc (x) − βc (y)| = |βc (x1 ) − βc (y1 )| = d(x1 , y1 ). § 3. Reconstruction of a Metric on a Virtually Higher-Rank Geodesic By Lemma 2.3, if an r-sequence defines a higher-rank geodesic a in a metric d then the geodesic a defined by this sequence has higher rank in an arbitrary V -metric d . In this section, we prove that a and a coincide pointwise and we have d = d along a. 3.1. Tapes. Definition 3.1. We say that a collection of 4p (p ∈ N) parallelly equivalent r-sequences {xi,j;z }z∈Z ,
i = 0, 3, j = 1, p,
(3.1)
forms a p-tape if the 4p + 4 points xi,1,0 , . . . , xi,p,0 , i = 0, 3,
x0,1,2p−1 , x2,p,1−2p , x3,p−1,1−2p , x3,p,1−2p ,
form the system of segments of r-sequences complementarily: ⎧ ⎪ ⎪ [x0,1,0 , x1,1,0 , x2,1,0 , x3,1,0 ]r , ⎪ ⎪ ⎪ ................................. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [x0,p,0 , x1,p,0 , x2,p,0 , x3,p,0 ]r , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ [x0,2,0 , x1,1,0 , x2,p,1−2p , x3,p−1,1−2p ]r , [x0,3,0 , x1,2,0 , x2,1,0 , x3,p,1−2p ]r , ⎪ ⎪ ⎪ ⎪ [x0,4,0 , x1,3,0 , x2,2,0 , x3,1,0 ]r , ⎪ ⎪ ⎪ ⎪ ⎪ .................................. ⎪ ⎪ ⎪ ⎪ ⎪ [x0,p,0 , x1,p−1,0 , x2,p−2,0 , x3,p−3,0 ]r , ⎪ ⎪ ⎪ ⎩ [x0,1,2p−1 , x1,p,0 , x2,p−1,0 , x3,p−2,0 ]r .
(3.2)
7
The definition of p-tape is given in terms of r-sequences and does not depend on the choice of a V metric. Fig. 1 represents a fragment of a p-tape. It shows only part of the points of the r-sequences that form the p-tape. In the interval between the points x1,1,0 and x1,1,2p−1 belonging to the same r-sequence {x1,1,z }z∈Z , there are 2(p − 1) points of this r-sequence, which divide the segment of the corresponding geodesic into 2p − 1 equal segments. This segment also contains p − 1 points of the form x1,j,0 , j = 2, p, which divide it into p equal segments. Thus all points of r-sequences {x1,j,z }z∈Z that are in the segment [x1,1,0 x1,1,2p−1 ] divide it into p(2p − 1) equal segments, and the segment [x1,1,0 x1,1,1 ] is divided by x1,p,3−2p , x1,p−1,5−2p , . . . , x1,2,−1 into p segments of length 1/p. x3, p, 1−2p x r`` 2, 1, 0
x
x
3, 1, 0 3, p−1, 0 r`` r`` r ` ` ``r ``r` ``r` ``r` r`` ```r ```r ```r ``r` ` ` ` ```r ```r ```r ```r r
x0, 1, 0 x1, 1, 0
x0, 2, 0
x0, p, 0
x1, p, 0 x 0, 1, 2p−1
Fig. 1. p-Tape. The plan for studying a higher-rank geodesic a is as follows: the curve a : R → X, defined by the r-sequence {xz = a(z)}z∈Z , is included in a flat strip. It can lie inside a strip or be the boundary geodesic without getting into a flat strip. If a is an interior curve of a flat strip F then, if P ∈ N large enough, then for every p > P , there is a p-tape in the strip defined by a family of r-sequences of the kind of (3.1) such that xz = x1,1,z for all z ∈ Z. The points of r-sequences x1,j,z for all such p-tapes for all p > P fill the set of points of the geodesic with rational values of the parameter. It suffices now to prove that the containment of points in a may be reconstructed from V . For points with rational values of the parameter, this follows directly from the construction of tapes; for irrational values, passage to the limit is possible by the topological equivalence of V -metrics. We then apply passage to the limit to geodesics lying on the boundaries of flat strips but not getting inside. 3.2. Reconstruction of a metric of a higher-rank geodesic. Here we carry out the above plan. Lemma 3.1. Suppose that r-sequences (3.1) form a p-tape. Then the r-sequences {x1,j,z }z∈Z and {x2,j,z }z∈Z are included in a flat strip. Proof. Let a p-tape be defined by (3.1). Consider a flat strip F included between the parallel geodesics a and a that include r-sequences {x0,1,z }z∈Z and {x3,1,z }z∈Z respectively. Then this strip also includes the sequences {x1,1,z }z∈Z and {x2,1,z }z∈Z . Prove that F in fact contains all points in the lemma. Consider the points x0,2,0 , x1,1,0 , x2,1,0 , x3,1,0 , x2,2,0 , x1,2,0 .
(3.3)
System (3.2) yields segments of r-sequences [x0,2,0 , x1,2,0 , x2,2,0 ]r ,
[x1,p,1−2p , x2,p,1−2p , x3,p,1−2p ]r .
Since the metric function is convex in a nonpositively curved space, the function d2,1 (t) := |xt,2,0 xt+1,1,0 |, where xt,j,0 = cj (t) and cj : [0, 2] → X, j = 1, 2, are the natural parametrizations of the segments [x0,j,0 x1,j,0 ], is convex. But d2,1 (0) = d2,1 (1) = d2,1 (2) = 1, and hence d2,1 (t) = 1 for all t ∈ [0, 2] and the points (3.3) belong to the planar parallelogram x0,2,0 x1,1,0 x3,1,0 x2,2,0 isometrically embedded in X. 8
Similarly, the six points x2,1,0 , x3,1,0 , x2,2,0 , x1,3,0 , x0,3,0 , x1,2,0 belong to the planar parallelogram x2,1,0 x3,1,0 x1,3,0 x0,3,0 isometrically embedded in X and |x2,1,0 x2,2,0 | = |x1,2,0 x1,3,0 |. The convex quadrangle x1,1,0 x2,1,0 x2,2,0 x1,3,0 is obtained as the union of rhombi x1,1,0 x2,1,0 x2,2,0 x1,2,0 ,
x1,2,0 x2,1,0 x2,2,0 x1,3,0
having the common triangle x2,1,0 x2,2,0 x1,2,0 . Therefore, it is isometric to a planar trapezium. The point x1,2,0 lies on the segment [x1,1,0 x1,3,0 ] and is its midpoint. Continuing similar arguments, we see that the segment [x1,1,0 x1,1,2p−1 ] is divided by the points x1,j,0 , j = 2, p, into p parts equal to the segment [x1,1,0 x1,2,0 ]. Thus, the starting points of all r-sequences of the form {x1,j,z }z∈Z lie on the boundary a of F . Hence, a includes all these r-sequences. The curve a in turn includes all r-sequences of the form {x2,j,z }z∈Z . Remark 3.1. In [2], the case was considered when the space does not admit overlapping minimizers. In this case, no flat strip overlap is possible either, from which it follows that an entire tape is included in one flat strip. In general, the “petals” of a tape that contain its extreme r-sequences can bend and pass from one strip onto another. Lemma 3.1 shows that the central part of a tape behaves in a controllable way. Lemma 3.2. The distance between parallel geodesics in X that include the r-sequences {x0,j,z }z∈Z and {x3,k,z }z∈Z of the given tape does not depend on j, k and is equal to √ 3 4p − 1 . s(p) = 2p Proof. The distance in question is equal to the three times distance between the curves a and a in Lemma 3.1. Since each flat strip in a CAT(0)-space is isometric to a strip on the Euclidean plane, for calculating this distance, it suffices to find the width of a p-tape on the Euclidean plane. This is elementary. Corollary 3.1. If a geodesic a in X lies inside a flat strip then there exists P > 0 such that, for all p > P , the flat strip includes the p-tape generated by a family of sequences of the form (3.1); moreover, {x1,j,z } ⊂ a. Lemma 3.3. Suppose that a higher-rank geodesic a : R → X in a CAT(0)-space X lies inside a flat strip F and contains an r-sequence {xz }z∈Z , x0 = a(0). Then, for every rational q := m n and for p divisible by n, every p-tape such that xz = x0,1,z contains the point a(q) = x0,j,z for j := n + 1 − m and z := q + 1 − 2k(n − m ), where q := [q] is the integer part and mn := {q} is the fractional part of q. Proof proceeds with direct calculation for a p-tape on the plane. Thus, if p is divisible by n then every p-tape constructed from the points xz = x1,1,z of a given r-sequence {xz }z∈Z contains the points a( m n ) as points with the multi-index defined by p. Now, we turn to proving the main theorem for higher-rank geodesics. 9
Theorem 3.1. Let a : R → X be a higher-rank geodesic in the space (X, d) that includes the r-sequence {xz }, z ∈ Z. Then, for every V -metric d on X, the curve a is a geodesic and d = d along it. Proof. As mentioned, the diagonal tube V determines the property of an r-sequence {xz }z∈Z to define a higher-rank geodesic. By Lemma 3.3, if a lies inside a flat strip then all V -metrics have the same set of rational points, i.e., points of the form a(q), where q ∈ Q. Let t be an irrational number and let tn , tn , n ∈ N, be a sequence of its rational approximations with deficiency and excess respectively. Then ∞ ∞ B(a((tn − 1), 1) ∩ B(a(tn + 1), 1) a(t) = n=1
n=1
and a(t) is uniquely determined from V . Assume now that a lies on the boundary of a flat strip F and is not inside any flat strip. Then, arbitrarily close to a, there is a geodesic a parallel to a along which d = d . The topological equivalence of V -metrics implies the equality d = d along a as well. Corollary 3.2. Let a be a virtually higher-rank geodesic in X. Then the equality d = d holds along a for an arbitrary metric d realizing V . Proof is carried out by consecutively applying Lemma 2.4 for the horospherical transfer from a geodesic to a geodesic asymptotic to it. § 4. Reconstruction of a Metric on a Rank-One Geodesic 4.1. Scissors. Definition 4.1. We say that geodesics a, b, c, d : (−∞, +∞) → X in a nonpositively curved space X form scissors with center a point x ∈ X if • a(−∞) = b(−∞); • a(+∞) = c(+∞); • c(−∞) = d(−∞); • b(+∞) = d(+∞); • b ∩ c = x. Denote such a configuration by a, b, c, d; x (Fig. 2). Geodesics a and d are called the bases of the scissors. The first base is said to be lower and the second, upper. The bases may or may not pass through the center of the scissors. In [2], the case was considered when the space X does not admit overlapping geodesics. In this case, the bases of scissors do not contain their center. The four infinite points at the endpoints of the geodesics a, b, c, d generate the four classes of Busemann functions represented by functions βa(±∞) and βd(±∞) vanishing at x: βa(±∞) (x) = βd(±∞) (x) = 0. d
((( hhhh ((( ( x( ( hhh ( r h ((h hhhh c b (((( ( hhhh ( (( a hhh
Fig. 2. Scissors a, b, c, d; x. Closely connected with scissors is the shift transformation of their lower base a, which leads to solving the main problem. It is defined as the composition of the following axial isometries. Let Rac be the horospherical transfer from a to c generated by the Busemann function βa(+∞) : an arbitrary point m ∈ a goes to the unique point m = Rac (m) ∈ c satisfying βa(+∞) (m ) = βa(+∞) (m). Similarly, we define transfers Rcd , Rdb , and Rba , which are isometric mappings of the corresponding geodesics by Busemann functions with centers at the common infinite points. 10
Definition 4.2. The shift T is the composition T := Rba ◦ Rdb ◦ Rcd ◦ Rac : a → a. Obviously, T is an isometry of the geodesic a preserving its direction. The displacement of T , i.e., the difference βa(−∞) (T (m)) − βa(−∞) (m), independent of m ∈ a, is denoted by δT . The quantity δT admits the following description: Let βa− , βa+ , βd− , and βd+ be Busemann functions with centers at a(±∞) and d(±∞) respectively such that there exist p ∈ a and q ∈ d satisfying βa− (p) = βa+ (p) = 0 and βd− (q) = βd+ (q) = 0. Theorem 4.1. We have δT = βa− (x) + βa+ (x) + βd− (x) + βd+ (x) ≥ 0.
(4.1)
Furthermore, if a is a strictly rank-one geodesic and a ∩ d = ∅ then δT > 0. Proof. Note that the sums βa− (x) + βa+ (x) and βd− (x) + βd+ (x) are independent of the choice of p ∈ a and q ∈ d since, say, in passing from p to a point p ∈ a, the functions βa− and βa+ change by constants equal to βa− (p ) and βa+ (p ) respectively; moreover, βa− (p ) = −βa+ (p ). We have δT = t − t, where a(t ) = T (a(t)). If d(s) = Rcd ◦Rac (a(0)) then d(s+t) = Rcd ◦Rac (a(t)) for all t ∈ R. Similarly, if a(t) = Rba ◦Rdb (d(0)) then a(t + s) = Rba ◦ Rdb (d(s)) for all s ∈ R. −1 (x) and q = d(0) = R−1 (x). Then β (x) = β (p) = 0, β (x) = β (q) = 0, Put p = a(0) = Rac a+ a+ d+ d+ db and T (p) = Rba ◦ Rdb ◦ Rcd (x) = Rba ◦ Rdb (d(s)), where s = βd− (x). Furthermore, T (p) = a(βd− (x) + t), where t = βa− (x). Thus, the shift of the point p ∈ a and hence of every point of a is equal to δT = βa− (x) + βd− (x) − 0 = βa− (x) + βa+ (x) + βd− (x) + βd+ (x). Since x is in the intersections of horoballs H B(a(+∞), x) ∩ H B(a(−∞), x) and H B(d(+∞), x) ∩ H B(d(−∞), x), it follows that δT ≥ 0. If a ∩ d = ∅ then we may assume that x ∈ / a. In this case, bd− (x) + bd+ (x) ≥ 0 and ba− (x) + ba+ (x) > 0. 4.2. Shadows. Definition 4.3. By the full shadow of a point x0 with respect to a point y ∈ X \ {x0 } we mean the set Shadowy (x0 ) := {z ∈ X | ∃[yz]x0 ∈ [yz]}. Here the existence assumption is necessary only if both points y, z are infinite: y, z ∈ ∂∞ X. By the spherical shadow of a point x0 of radius ρ > 0 with respect to a point y ∈ X we mean the intersection Shadowy (x0 , ρ) of its full shadow Shadowy (x0 ) with the sphere S(x0 , ρ). In particular, if ρ = +∞ then Shadowy (x0 , +∞) := ∂∞ (Shadowy (x0 )) := Shadowy (x0 ) ∩ ∂∞ X. The following properties of shadows are obvious: (1) The sets Shadowy (x0 ) ∪ {x0 } and Shadowy (x0 , ρ) are closed in X for all ρ > 0. (2) Shadowy (x0 ) = ρ>0 Shadowy (x0 , ρ) \ {x0 }. (3) If y ∈ X then Shadowy (x0 , ρ) = S(y, |x0 y| + ρ) ∩ S(x0 , ρ). (4) If y ∈ ∂∞ X then Shadowy (x0 , ρ) = (H S y,ρ ) ∩ S(x0 , ρ); here H S y,ρ stands for the horoball H S y,z , where z ∈ X is a point such that if by is a Busemann function with center y then by (z) − by (x0 ) = ρ. (5) If ∠x0 (y, z) = 0 then Shadowy (x0 ) = Shadowz (x0 ). Property (5) implies 11
Corollary 4.1. If the direction at the point c(0) = x0 of the ray c : [0, ∞) → X that contains a point y has a unique opposite direction then Shadowz (x0 ) = Shadowz (x0 ) for arbitrary z , z ∈ Shadowy (x0 ). Given a sufficiently small ε > 0, by the ε-neighborhood Nε (Shadowy (x0 , ρ)) of a spherical shadow Shadowy (x0 , ρ) with ρ < +∞ we mean its ε-neighborhood on the sphere S(y, |x0 y| + ρ) if y ∈ X or on the horosphere H S y,ρ if y ∈ ∂∞ X. Theorem 4.2. For all x0 ∈ X, y ∈ X \ {x0 } and numbers 0 < ρ < +∞ and ε > 0, there exists δ > 0 such that the inclusion Shadowy (x1 , ρ) ⊂ Nε (Shadowy (x0 , ρ)) holds for every point x1 ∈ B(x0 , δ) for which |yx1 | = |yx0 | (or by (x1 ) = by (x0 ) if y ∈ ∂∞ X). Proof. We prove the theorem for the case of y ∈ X by contradiction. The case of y ∈ ∂∞ X is considered likewise. Suppose that, for every δ > 0, there exist xδ ∈ B(x0 , δ) ∩ S(y, |yx0 |) and zδ ∈ S(y, |yx0 |+ρ)\Nε (Shadowy (x0 , ρ)) satisfying xδ ∈ [yzδ ]. Choose a sequence δn → 0 and the corresponding sequences xδn and zδn . We have xδn → x0 . By finite compactness of X, we can extract a convergent subsequence from zδn . Assume that zδn itself converges to a point z ∈ S(y, |yx0 | + ρ). In the natural parametrization of the segment γ = [yz], we have (4.2) |x0 γ(|yx0 |)| ≤ |x0 xδn | + |xδn γ(|yx0 |) ≤ δn + |zδn z|. The right-hand side of (4.2) vanishes as n → ∞; consequently, the constant on the left-hand side is 0. Hence, z ∈ Shadowy (x0 , ρ) and the points zn belong to Nε (Shadowy (x0 , ρ)) for sufficiently large n, which contradicts the choice of zn . Given a set V ⊂ ∂∞ X and numbers K, ε > 0, consider the (y, K, ε)-neighborhood V , i.e., the set Ny,K,ε (V ) := {ζ ∈ ∂∞ X | ∃ξ ∈ V ζ ∈ U (ξ, y, K, ε)}, where U (ξ, x0 , K, δ) := {η ∈ ∂∞ X | |c(K)d(K)| < ε, c = [y, ξ]; d = [y, η]}. The following assertion is but a reformulation of Theorem 4.2 for y ∈ X. Corollary 4.2. For every neighborhood Ny,K,ε (∂∞ (Shadowy (x0 ))) of the shadow at infinity ∂∞ (Shadowy (x0 )) and every y ∈ X, there exists δ > 0 such that the inclusion ∂∞ (Shadowy (x1 )) ⊂ Ny,K,ε (∂∞ (Shadowy (x0 ))) holds for every x1 ∈ B(x0 , δ) with |yx1 | = |yx0 |. 4.3. Geometry of the ideal boundary of a CAT(0)-space. In this subsection, we recall some available facts of the asymptotic geometry of nonpositively curved space (see [11] for details). Observe first that the cone topology of the ideal boundary admits the following description: For δ > 0 fixed and a distinguished point o ∈ X, a base of neighborhoods of a point ξ ∈ ∂∞ X is formed by the family of sets (4.3) Bo,ξ := {Uδ,t (o, ξ) | t > 0}. Here Uδ,t (o, ξ) := {η ∈ ∂∞ X | |c(t)d(t)| < δ}, where c, d : [0, +∞) → X are rays starting from o in the directions ξ, η ∈ ∂∞ X respectively. Apart from the cone topology, the boundary of X admits a canonic angle metric and the so-called Tits metric. For ξ, η ∈ ∂∞ X, the angle distance between them is ∠(ξ, η) := sup{∠x (ξ, η) | x ∈ X}. The Tits metric is the intrinsic metric generated by the angle metric. These two metrics are equivalent, i.e., generate the same topology on ∂∞ X. 12
Proposition 4.1 [11, Proposition 9.5]. The angle metric ∠ is lower semicontinuous in the cone topology as a function (ξ, η) → ∠(ξ, η): for every ε > 0, there exist a neighborhood U of ξ and a neighborhood V of η for which the containments ξ ∈ U and η ∈ V imply that ∠(ξ , η ) > ∠(ξ, η) − ε. As a consequence, the Tits metric is also lower semicontinuous in the cone topology. Proposition 4.2 [11, Proposition 9.21]. Let ξ0 , ξ1 be two distinct points of the ideal boundary ∂∞ X. 1. If Td(ξ0 , ξ1 ) > π then there exists a geodesic c : R → X with c(+∞) = ξ0 and c(−∞) = ξ1 . 2. If there is no geodesic c : R → X with c(+∞) = ξ0 and c(−∞) = ξ1 then Td(ξ0 , ξ1 ) = ∠(ξ0 , ξ1 ) and there exists a geodesic segment in the Tits metric between ξ0 with ξ1 . 3. Given a geodesic c : R → X, we have Td(c(−∞), c(+∞)) ≥ π, and equality here holds if and only if c lies on the boundary of a Euclidean half-plane isometrically embedded in X. 4. If the diameter of the boundary in the Tits metric is equal to π then every geodesic in X bounds an isometrically embedded Euclidean half-plane. 4.4. Points of a geodesic with a unique opposite direction. Let Σx X be the space of directions at a point x ∈ X. Directions ξ, η ∈ Σx X are called mutually opposite if ∠x (ξ, η) = π. If the space is geodesically complete then two directions ξ, η ∈ Σx X are mutually opposite if and only if there exists a geodesic passing through x so that its positive direction at x is ξ and its negative direction at x is η. Given a geodesic a in a CAT(0)-space X, denote by ω + (a) the set of points x ∈ a at which the positive direction of a has more than one opposite direction. Similarly, ω − (a) is the set of points on a at which the negative direction of a has more than one opposite direction. Theorem 4.3. The sets ω + (a) and ω − (a) are at most countable. + + Proof. In case φ > 0, consider the set Ω+ φ (a) ⊂ ω (a) defined as follows: x ∈ Ωφ (a) if and only if there exists a direction ζ ∈ Σx X opposite to the direction of the ray [xa(+∞)] satisfying ∠x (ζ, a(−∞)) > φ. Prove that the intersection of Ω+ φ (a) with every segment [xy] ⊂ a is finite. Indeed, supposing that a segment [xy] ⊂ a contains an infinite sequence of points {a(tn )}∞ n=1 ⊂ + Ωφ (a), we arrive at the following contradiction with local compactness of X. Assume that x = a(0) and y = a(−L), where L = |xy|. Given a(tn ), find a point zn ∈ S(x, 2L) with a(tn ) ∈ [xzn ] and ∠a(tn ) (a(−∞), zn ) > φ. Now, we see that |zn zk | > 2L sin φ2 for all n = k. Hence, the sequence {zn }∞ n=1 has no Cauchy subsequence, and the sphere S(x, 2L) is not compact. + + + Since ω + (a) = φ>0 Ω+ φ (a) and Ωφ (a) ⊂ Ωψ (a) for ψ < φ, the theorem holds for ω (a). The case of − ω (a) is considered similarly. 4.5. Existence of scissors. In this subsection, we prove the following theorem on existence of scissors: Theorem 4.4. Let a : (−∞, +∞) → X be a strictly rank-one geodesic and let x0 ∈ a be a point at which both directions of a have unique opposite directions. Then there exists a geodesic a passing through a (0) = x0 so that ∠x0 (a(+∞), a (+∞)) = 0, with the following property: For every neighborhood U of the triple (a (+∞), a (−∞), x0 ) ∈ ∂∞ X × ∂∞ X × X, there exist a triple (ξ, η, x) ∈ U with x = x0 and scissors a , b , c , d ; x satisfying b = [a (−∞)ξ], c = [ηa (+∞)], and d = [ηξ]. Proof. Observe that the assumption on the rank of a implies that every geodesic a satisfying a (+∞) ∈ ∂∞ (Shadowa(−∞) (x0 ))
or a (−∞) ∈ ∂∞ (Shadowa(+∞) (x0 ))
has rank 1. We first show that there exist scissors with lower base a and center x that is arbitrarily close to x0 . In light of the above remark, this argument is in particular applicable to the geodesic a passing through x0 in the same direction as a. 13
Given ρ > 0, consider the points y = a(−ρ) and y = a(ρ). We have ∂∞ (Shadowa(−∞) (x0 )) = ∂∞ (Shadowy (x0 )), ∂∞ (Shadowa(+∞) (x0 )) = ∂∞ (Shadowy (x0 )). The lower semicontinuous function Td : ∂∞ X × ∂∞ X → R+ ∪ {+∞} attains its minimum on the compact set Q = ∂∞ (Shadowy (x0 )) × ∂∞ (Shadowy (x0 )). This, together with the assumption on the rank of a, yields the inequality min(Td)|Q > π.
(4.4)
Furthermore, for some K > ρ and ε > 0, there are neighborhoods N := Ny ,K,ε (∂∞ (Shadowy (x0 ))),
N
:= Ny ,K,ε (∂∞ (Shadowy (x0 )))
of the shadows at infinity of x0 satisfying inf{Td(ξ, η) | (ξ, η) ∈ N × N } > π.
(4.5)
Choose a δ1 -neighborhood B(x0 , δ1 ) of x0 defined by Corollary 4.2 from the neighborhoods Ny ,K,ε/2 (∂(Shadowy (x0 , ρ))),
Ny ,K,ε/2 (∂(Shadowy (x0 , ρ))).
Suppose also that Nε/2 (∂(Shadowa(−∞) (x0 , ρ))) and Nε/2 (∂(Shadowa(+∞) (x0 , ρ))) are the 2ε -neighborhoods of the boundaries of the shadows of x0 with respect to a(−∞) and a(+∞). By Theorem 4.2, there exists a δ2 -neighborhood B(x0 , δ2 ) of x0 in which the inclusions Shadowa(−∞) (x , ρ) ⊂ Nε/2 (∂(Shadowa(−∞) (x0 , ρ))), Shadowa(+∞) (x , ρ) ⊂ Nε/2 (∂(Shadowa(+∞) (x0 , ρ))) hold at every x ∈ B(x0 , δ2 ). Put δ0 := min{δ1 , δ2 }. Then, given a point x ∈ Uδ0 (x0 ) and geodesics b and c meeting the conditions • b(0) = c(0) = x, • b(−∞) = a(−∞), • c(+∞) = a(+∞), we have the containments b(ρ) ∈ Nε/2 (∂(Shadowa(−∞) (x0 , ρ))), (4.6) c(−ρ) ∈ Nε/2 (∂(Shadowa(+∞) (x0 , ρ))). Prove that b(+∞) ∈ N and
c(−∞) ∈ N .
(4.7) (4.8)
The ray γ = [y b(+∞)] with the natural parametrization γ : [0, +∞) → X and the geodesic a passing through x0 so that a (+∞) ∈ ∂∞ (Shadowy (x0 )) meet the inequality |γ(2ρ)a (ρ)| ≤ |γ(2ρ)b(ρ)| + |b(ρ)a (ρ)|. 14
The first summand here is estimated as follows: ε |γ(2ρ)b(ρ)| ≤ |γ(0)b(−ρ)| = |a(−ρ)b(−ρ)| ≤ |a(0)b(0)| < . 2 By (4.6), a can be chosen so that the second summand also satisfies the estimate ε |b(ρ)a (ρ)| < . 2 Finally,
|γ(2ρ)a (ρ)| < ε,
which proves (4.7). The case of (4.8) is considered similarly. Thus, by (4.5), there exists a geodesic d in X joining c(−∞) and b(+∞) that forms scissors
a, b, c, d; x. Now, taking a sequence δn → 0, for each δn , construct scissors an , bn , cn , dn ; xn with |x0 xn | < δn . Choose limit points ξ, η ∈ ∂∞ X for the sequences bn (+∞) and cn (−∞) respectively. Then ξ ∈ ∂∞ (Shadowa(−∞) (x0 )),
η ∈ ∂∞ (Shadowa(+∞) (x0 )).
Hence, ξ and η can be joined by a geodesic a = [ηξ] in X so that a (0) = x0 and, for every δ > 0, there exist scissors a , b , c , d ; x with lower base a and center x such that |xx0 | < δ. Note that x can always be chosen different from x0 and not belonging to a . We are left with proving that such scissors can be chosen so that b (+∞) is arbitrarily close to ξ and c (−∞), to η in the cone topology on ∂∞ X. This can be done by repeating the above argument for neighborhoods of the points a (±∞) instead of neighborhoods of the shadows at infinity of x0 . The construction of a guarantees that the sets
[ya (−∞)], C−∞ (B(a (K), ε)) = y∈B(a (K),ε)
C+∞ (B(a (−K), ε)) =
[za (+∞)]
z∈B(a (−K),ε)
intersect for all ε, K > 0 and, moreover, C−∞ (Nε (a (K))) ∩ C+∞ (Nε (a (−K))) ∩ X \ (Shadowy (x0 ) ∪ Shadowy (x0 ) ∪ {x0 }) ∩ B(x0 , ε) = ∅. This suffices to carry out the desired construction. 4.6. Continuity of the displacement. In this subsection, we prove continuity of the displacement function δ as a function on an appropriate subset in ∂∞ X × ∂∞ X × X. The proof here is much simpler than that of a similar theorem in [2]. Let a : R → X be a strictly rank-one geodesic. Denote by Z(a) ⊂ ∂∞ X × ∂∞ X × X the subset consisting of triples (ξ, η, x) ∈ ∂∞ X ×∂∞ X ×X such that there exist scissors a, b, c, d; x with b(+∞) = ξ and c(−∞) = η. Theorem 4.5. The displacement function δ is continuous on Z(a). Proof. Use (4.1). Suppose that we have a triple (ξ0 , η0 , x0 ) ∈ Z(a). The point x0 is the center of the scissors a, b0 , c0 , d0 ; x0 , where b0 (+∞) = ξ0 and c0 (−∞) = η0 . Suppose that ε > 0. Then, first, continuity of the Busemann functions ba− and ba+ implies that there exists a number σ1 such that if a point x ∈ X meets the inequality |x0 x | < σ1 then |ba+ (x ) + ba− (x ) − ba+ (x0 ) − ba− (x0 )| < ε/3.
(4.9) 15
Proceed with using the coincidence of the geodesic and metric boundaries of X. In other words, every neighborhood U of an ideal geodesic point ξ ∈ ∂g X is a neighborhood of the ideal metric point defined by the Busemann function βξ and vice versa. Choose a neighborhood V of x0 with compact closure in which the values of the Busemann function bd± differ from bd± (x0 ) by at most ε/6. Denote by 1 1 U± (V ) := C V, b± (x0 ) − ε, b± (x0 ) + ε 6 6 the neighborhoods of b± in C(X) containing the functions taking values on V different from those of b± at most by ε/6, and denote by U± := U± / consts ∩∂m X the neighborhoods generated by them in ∂m X = ∂g X. Coincidence of the boundaries means that if a ray d is such that d (+∞) ∈ U+ then the Busemann function βd generated by it in V meets the inequality |bd+ (x0 ) − βd (x ) − const | < ε/3 at every point x with some constant, and if a ray d is such that d (+∞) ∈ U− then in V we have |bd− (x0 ) − βd (x ) − const | < ε/3. Since all Busemann functions with common center differ by constants, it follows that the constants in the preceding inequalities can be taken zero. Put U = (U+ × U− × V ) ∩ Z. If (ξ , η , x ) ∈ U is a triple generating scissors a, b , c , d ; x with displacement δ then we have |δ − δ| = |(ba− (x0 ) + ba+ (x0 ) + bd− (x0 ) + bd+ (x0 )) −(ba− (x ) + ba+ (x ) + bd − (x ) + bd + (x ))| < ε.
Theorem 4.6. Let x0 ∈ a be a point on a strictly rank-one geodesic a. If (ξ, η, x) = (b(+∞), c(−∞), x) ⊂ Z tends to (a(+∞), a(−∞), x0 ) in the product topology on ∂∞ X × ∂∞ X × X then δ(ξ, η, x) → 0. Proof. Define the “closed ” scissors a, a, a, a; x0 as the set of four copies of a and the center x0 . Here it is natural to consider the identity mapping T = ida as a shift satisfying δT = 0. Moreover, the function δ remains continuous if extended to Z(a) ∪ {(a(+∞), a(−∞), x) | x ∈ a} by the equality δ(a(+∞), a(−∞), x) = 0 for x ∈ a. The following assertion plays a key role in the remainder of the proof of Theorem 1.4. Corollary 4.3. Let a be a strictly rank-one geodesic. Then there exist a point x0 ∈ a, a geodesic a joined with a by an asymptotic chain, and a number Δ > 0 such that, for every δ ∈ (0, Δ), there exist scissors a , b, c, d; x for which the displacement of T is equal to δT = δ. 4.7. Reconstruction of a metric on a rank-one geodesic. The process of reconstruction of a metric on a strictly rank-one geodesic corresponds to the procedure of [2] completely. Here we give proofs in terms of V -metrics. Lemma 4.1. Suppose that x ∈ / a and ξ, η ∈ ∂∞ X. If a triple (ξ, η, x) belongs to Z(a) in the metric d then (ξ, η, x) ∈ Z(a) in every V -metric d on X. Proof. If two complete geodesics γ and γ defined by r-sequences {xz }z∈Z are asymptotic in the metric d then they are asymptotic in the same direction with respect to an arbitrary metric d that realizes V . The lemma follows from the fact that the notion of scissors bases only on asymptoticity of the geodesics. 16
Lemma 4.2. Suppose that a is a strictly rank-one geodesic. Then, given scissors a, b, c, d; x, the image T (m) ∈ a of an arbitrary point m ∈ a under the shift T does not depend on the choice of a V -metric on X. Proof. By the assumption on the rank of a, all four geodesics in the scissors are of rank one. Hence, they are all geodesic in an arbitrary V -metric d . The transformations Rab etc., yielding T , are defined for the point m by using intersections of the above geodesics with horospheres. Since the set of horospheres does not depend on the choice of a V -metric, the image T (m) is defined independently of d . Theorem 4.7. Let a be a strictly rank-one geodesic in (X, d). Then it is a geodesic in an arbitrary V -metric d and d = d along a. Proof. The first assertion of the theorem is a corollary to the assumption on the rank of the geodesic a : R → X. By Corollary 4.3, there exists N ∈ N such that, for all n > N , there exist scissors with shift Tn and displacement δTn = 1/n. Put x0 = a(0), x = Tn (x), and x1 = (Tn )n (x) = a(1). By (4.1), in a V -metric, δT does not depend on the choice of the initial point. Therefore, d(x0 , x ) = d (x0 , x ) = 1/n. Moreover, if for points x, y ∈ a the distance d(x, y) is rational then d(x, y) = d (x, y). Passage to the limit similar to that in the proof of Theorem 3.1 leads to the equality d = d for arbitrary points of a. Now we are in a position to prove the main assertion of the article. Proof of Theorem 1.4. Equivalence of items (1)–(3) was proven in Lemmas 2.1 and 2.2. They all are obvious from (4). We are left with proving that (4) follows from (1)–(3), i.e., that d is a unique metric on X realizing V . Let d be a V -metric on X. Consider arbitrary points x, y ∈ X. The segment [xy] is included in a complete geodesic a. Such a geodesic can be nonunique, but it suffices to consider any of them. If a has higher or virtually higher rank then d(x, y) = d (x, y) by Theorem 3.1 or Corollary 3.2. If a has strictly higher rank then the distances coincide in view of Theorem 4.7. References 1. Berestovskiˇı V. N., “Isometries in Aleksandrov spaces of curvature bounded above,” Illinois J. Math., 46, No. 2, 645–656 (2002). 2. Andreev P. D., “Recovery of a metric of the CAT(0)-space by a diagonal tube,” Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 299, 5–29 (2003). 3. Berestovski˘ı V. N., “Busemann spaces of curvature bounded above in the sense of Alexandrov,” Algebra i Analiz, 46, No. 2, 645–656 (2002). 4. Beckman F. S. and Quarles D. A. Jr., “On isometries of Euclidean spaces,” Proc. Amer. Math. Soc., 4, 810–815 (1953). 5. Alexandrov A. D., “On congruence-preserving mappings,” Dokl. Akad. Nauk SSSR, 211, No. 6, 1257–1260 (1973). 6. Kuz minykh A. V., “Mappings preserving unit distances,” Sibirsk. Mat. Zh., 29, No. 3, 597–602 (1979). 7. Tyszka A., “Discrete versions of the Beckman–Quarles theorem,” Aequationes Math., 59, 124–133 (2000). 8. Tyszka A., The Beckman–Quarles Theorem for Continuous Mappings from Cn to Cn [Preprint / ArXiv:math.MG / 0406093] (2004). 9. Alexandrov A. D., Berestovski˘ı V. N., and Nikolaev I. G., “Generalized Riemannian spaces,” Uspekhi Mat. Nauk, 41, No. 3, 3–44 (1986). 10. Ballmann W., Lectures on Spaces of Nonpositive Curvature, Birkha¨ user, Basel; Boston; Berlin (1995). 11. Bridson M. and Haefliger A., Metric Spaces of Nonpositive Curvature, Springer-Verlag, Berlin (1999) (Grundlehren der Math. Wiss.; 319). 12. Buyalo S. V., Lectures on Spaces of Curvature Bounded Above [Preprint], Univ. of Illinois, Urbana-Champaign (1994). 13. Alexander S. B. and Bishop R. L., “The Hadamard–Cartan theorem in locally convex spaces,” Enseign. Math., 36, 309–320 (1990). 14. Kapovich I. and Benakli N., “Boundaries of hyperbolic groups,” Contemp. Math., 296, 39–94 (2002). 15. Webster C. and Winchester A., Boundaries of Hyperbolic Metric Spaces [Preprint / ArXiv: math.MG / 0310101] (2003). 16. Ballmann W., Gromov M., and Schroeder V., Manifolds of Nonpositive Curvature, Birkha¨ user, Boston; Basel; Stuttgart (1985). P. D. Andreev Pomor State University, Arkhangel sk, Russia E-mail address:
[email protected] 17
