Noncritical maps on geodesically complete spaces with curvature bounded above

We define and study the regularity of distance maps on geodesically complete spaces with curvature bounded above. We prove that such a regular map is locally a Hurewicz fibration. This regularity can be regarded as a dual concept of Perelman's regularity in the geometry of Alexandrov spaces with curvature bounded below. As a corollary we obtain a sphere theorem for geodesically complete CAT(1) spaces.


Introduction
The notions of lower and upper curvature bounds for metric spaces were introduced by Alexandrov using triangle comparison.More specifically, a metric space has curvature ≥ κ (resp.≤ κ) if any small geodesic triangle is "thicker" (resp."thinner") than the geodesic triangle with the same sidelengths on the plane of constant curvature κ.Metric spaces with curvature bounded below and above are abbreviated as CBB spaces and CBA spaces, respectively.The properties of both curvature bounds are completely different in general.
CBB spaces are usually called Alexandrov spaces and play an essential role in the convergence theory of Riemannian manifolds with sectional curvature bounded below.The fundamental theory of finite-dimensional CBB spaces was developed by Burago, Gromov, and Perelman [2].They showed that topological dimension and Hausdorff dimension coincide for CBB spaces and that any CBB space contains an open dense Lipschitz manifold of its dimension.To prove these, they defined a strainer (a collection of points around a point satisfying an orthogonal condition) and studied the distance map from points of a strainer, namely a strainer map.Perelman [13], [14] then developed the structure theory of CBB spaces by introducing the regularity of distance maps more general than that of strainer maps, which extends the regularity of distance functions introduced by Grove and Shiohama [3] in the Riemannian setting.In particular, he proved that such a regular map is locally a bundle map and obtained a stratification of an arbitrary CBB space into topological manifolds.
On the other hand, the structure of CBA spaces is much more complicated ( [6]).For example, the Hausdorff dimension may be bigger than the topological dimension.To obtain some control, we assume the (local) geodesic completeness, that is, the extension property of geodesics.A separable, locally compact, locally geodesically complete CBA space is called a GCBA space.Lytchak and Nagano [8], [9] recently published the fundamental theory of GCBA spaces (note also that there was an unpublished work of Otsu and Tanoue [12]).Their results suggest that the geometry of GCBA spaces has many parallels with that of CBB spaces.They showed the coincidence of topological and Hausdorff dimensions for GCBA spaces and proved that any GCBA space can be stratified in a measure-theoretic sense so that each stratum contains an open dense Lipschitz manifold of its dimension.They also obtained a manifold recognition theorem for GCBA spaces.Their main technical tool is a strainer on a GCBA space, which can be seen as a dual of a strainer on a CBB space.In particular, they showed that any strainer map is locally a Hurewicz fibration (but not a bundle map; see [10, 2.7] for example).
In this paper we define and study the regularity of distance maps on GCBA spaces more general than that of strainer maps, which can be regarded as a dual concept of Perelman's regularity in CBB geometry.To define it, we introduce the following notation.Let Σ be a compact, geodesically complete CAT(1) space with diameter π (any space of directions of a GCBA space satisfies these conditions).Let U be a tiny ball in a GCBA space X, that is, a small metric ball where triangle comparison holds (see Sec. 3.1 for the precise definition).In this paper we usually work inside some tiny ball.For p, a ∈ U , we denote by Σ p the space of directions at p and by a ′ p ∈ Σ p (or simply a ′ if no confusion arises) the direction of the unique shortest path from p to a. Using the above notation, the definition of a strainer by 7.2] can be expressed as follows: Here the first inequality guarantees that there is an almost unique extension of the shortest path a i p.On the other hand, a strainer in a CBB space is defined by the inequalities any choice of directions), where the first inequality means that a i p is almost extendable as a shortest path.Note that shortest paths in CBB spaces are neither unique nor extendable in general, but do not branch, whereas shortest paths in GCBA spaces are unique and extendable at least in tiny balls, but may branch.
We now define the regularity of distance maps dealt with in this paper.Let ε and δ be small positive numbers such that δ ≪ ε, where the choice of δ depends only on the (local) dimension and ε (more precisely, it will be determined by the proof of each statement; see Sec. 2).Definition 1.1.Let U be a tiny ball in a GCBA space X and a + ε (for any choice of directions).In case Σ p is a unit sphere and a ′ i , b ′ are unique, both definitions coincide.Note that in this case a ′ i are linearly independent as vectors in Euclidean space.Remark 1.3.One can also define the ε-regularity by strengthening the condition (1) to |a ′ i a ′ j | < π/2 − ε, as Perelman did in [14], and simplify some of the proofs.The error δ is only used in Lemma 5.10.In case k = 1 there is no difference.
Note that being a noncritical point of f is an open condition by the upper semicontinuity of angle and the local geodesic completeness.It is easy to see that )-noncritical at p for some constant c > 0 (use the c-openness of f to find b ′ ; see [8, 8.2]).
Example 1.4.Let X be the Euclidean cone over the circle of length 2π + θ, which is a geodesically complete CAT(0) space for θ ≥ 0.Then, (1) if θ < π/4, there exists a noncritical map f : X → R 2 at the vertex o; (2) if π/4 ≤ θ < π, there exists a noncritical function f : Example 1.5.Let p be an isolated singularity of a GCBA space, that is, a nonmanifold point such that its punctured neighborhood is a manifold.Then |ξη| = π for any ξ, η ∈ Σ p .In particular, there exists no noncritical function at p. Indeed, if |ξη| < π, then Σ p is covered by the two open balls of radius π centered at ξ and η, both of which are contractible.Since Σ p is a homology manifold ( [9, 3.3, 3.4]), the same argument as in [9, 8.2] shows that Σ p has the homotopy type of a sphere.Therefore the theorem of Lytchak-Nagano [9, 1.1] implies that p is a manifold point, which is a contradiction.
We prove the following two theorems generalizing the results of Lytchak-Nagano for strainer maps.The first extends [8, 8.2, 11.2] and the second extends [9, 5.1].As before, U denotes a tiny ball in a GCBA space X.Let T p denote the tangent cone at p ∈ U .Note that the local dimension dim T p may not be constant.We denote by c(ε) a positive constant depending only on the (local) dimension and ε.
Theorem 1.7.Let f : U → R k be a noncritical map at p ∈ U , where k < dim T p .Then there exists an arbitrarily small contractible open neighborhood V of p such that f : V → f (V ) is a Hurewicz fibration with contractible fibers.
Remark 1.8.The above theorems hold for noncritical maps on CBB spaces (note that the local dimension is constant for CBB spaces).Moreover, any (ε, δ)noncritical map at p in a CBB space is a bundle map near p with conical fibers, provided that δ is sufficiently small compared to the volume of Σ p .See [13] for the details (cf.[14]).
As a corollary we obtain the following sphere theorem.For other sphere theorems, see [11] and the references therein.We may assume that the diameter of a geodesically complete CAT(1) space is exactly π by considering the π-truncated metric if necessary (see also Remark 5.12).
Remark 1.10.The above estimate is optimal in the following sense.Let T be a tripod, that is, a metric space consisting of three points with pairwise distance π.Let Σ be the spherical join between S n−1 and T , which is not homeomorphic to S n .We regard S n−1 and T as isometrically embedded in Σ. Choose {ξ i } n+1 i=1 ⊂ S n−1 such that |ξ i ξ j | < π/2 and η ∈ T .Then we have |ξ i η| = π/2 for all i.
Remark 1.11.Lytchak-Nagano [9, 1.5] also proved the following sphere theorem: if Σ is a compact, geodesically complete CAT(1) space with no tripods, then it is homeomorphic to a sphere.The author does not know whether the assumption of the above corollary implies the absence of tripods.
Remark 1.12.There is the CBB counterpart of the above corollary: if Σ is a CBB(1) space of dimension n and if {ξ i } n+1 i=1 and η in Σ satisfy |ξ i ξ j | > π/2 − δ and |ξ i η| > π/2 + ǫ, then Σ is bi-Lipschitz homeomorphic to S n .Moreover, if there exist {ξ i } k i=1 and η in Σ satisfying the same inequalities, where k ≤ n, then Σ is homeomorphic to a k-fold suspension ( [13, 4.5], [4,Theorem C]).This paper is organized as follows.In Sec. 2 we introduce some notation used in this paper.In Sec. 3 we give preliminaries on GCBA spaces, ε-open maps, and Hurewicz fibrations.In Sec. 4 we study the properties of the differential of a noncritical map.In Sec. 5 we first prove Theorem 1.6 by using the results of the previous section.We then construct a local neighborhood retraction to the fiber of a noncritical map to prove Theorem 1.7.Finally we give the proof of Corollary 1.9.

Notation and conventions
We will use the following standard notation.For points p, q in a metric space, |pq| denotes the distance between them.For r > 0, we denote by B(p, r) (resp.B(p, r)) the open (resp.closed) metric ball of radius r centered at p.The boundary ∂B(p, r) is defined by the difference B(p, r) \ B(p, r).
We will also use the following notation from [13].As in the introduction, ε and δ denote positive numbers such that δ ≪ ε.The choice of δ depends only on the (local) dimension, the upper curvature bound, and ε (the dependence on the upper curvature bound is not necessary if it is taken to be nonnegative).Whenever ε and δ appear in a statement, it means that the statement holds for a suitable choice of δ depending on ε, which will be determined by the proof.We denote by c(ε) various positive constants such that c(ε) ≪ ε, and by κ(δ) various positive functions such that κ(δ) → 0 as δ → 0. They also depend only on the (local) dimension, the upper curvature bound, and ε.In particular we may assume κ(δ) ≪ c(ε) by taking δ ≪ ε.Whenever c(ε) and κ(δ) appear in a statement, it means that the statement holds for some c(ε) and κ(δ) determined by the proof.

Preliminaries
3.1.GCBA spaces.Here we recall basic notions and facts about GCBA spaces.We refer the reader to [8] and [1] for more details.We assume all metric spaces are separable and locally compact unless otherwise stated.
Let κ ∈ R. We denote by S 2 κ the complete simply-connected surface of constant curvature κ and by D κ the diameter of S 2 κ .A complete metric space is called a CAT(κ) space if any two points with distance < D κ can be joined by a shortest path and if any geodesic triangle with perimeter < 2D κ is not thicker than the comparison triangle on S 2 κ .A metric space is called a CBA(κ) space if any point has a CAT(κ) neighborhood.For example, a complete Riemannian manifold is CBA(κ) if and only if its sectional curvature is bounded above by κ and is CAT(κ) if in addition its injectivity radius is bounded below by D κ .One can also construct numerous examples by Reshetnyak's gluing theorem: a gluing of two CBA(κ) (resp.CAT(κ)) spaces along their isometric convex subsets is again a CBA(κ) (resp.CAT(κ)) space.
Let X be a CBA(κ) space.A geodesic is a curve that is locally a shortest path.We say that X is locally geodesically complete if any geodesic can be extended to a geodesic beyond its endpoints, and that X is geodesically complete if the extension can be defined on R. For example, if a small punctured ball at each point of X is noncontractible, then X is locally geodesically complete.In particular, any homology manifold with a CBA metric is locally geodesically complete.If X is complete, then the local geodesic completeness is equivalent to the geodesic completeness.A separable, locally compact, locally geodesically complete CBA space is called a GCBA space.
Let X be a GCBA(κ) space.The angle between two shortest paths is defined by the limit of comparison angles.The space of directions at p, denoted by Σ p , is the set of the directions of shortest paths emanating from p equipped with the angle metric.Σ p is a compact, geodesically complete CAT(1) space with diameter π.By the local geodesic completeness, for any given direction, there exists a shortest path starting in that direction.Furthermore any direction has at least one opposite direction.The tangent cone T p at p is the Euclidean cone over Σ p .T p is isometric to the blow-up limit of X at p and is a geodesically complete CAT(0) space.
The dimension of a GCBA space is defined by the Hausdorff dimension, which coincides with the topological dimension.Note that the local dimension dim T p is finite and upper semicontinuous, but not necessarily constant.
We say that a metric ball U in X of radius r is tiny if the closed concentric ball of radius 10r is a compact CAT(κ) space and r < min{D κ /100, 1}.In this paper we usually work inside some tiny ball.Any two points in U are joined by a unique shortest path contained in U and any shortest path in U can be extended to a shortest path of length 9r.The angle is upper semicontinuous in U .For p, a ∈ U , we denote by a ′ p ∈ Σ p (or simply a ′ if no confusion arises) the direction of the unique shortest path from p to a. Let f = |a • |.Then f is convex on U and the directional derivative f ′ of f on Σ p is given by the first variation formula The directional derivative is defined as follows.Let f be a (locally Lipschitz) function defined on an open subset U of a GCBA space.For p ∈ U and ξ ∈ Σ p , let γ ξ (t) denote a shortest path starting at p in the direction ξ and parametrized by arclength.The directional derivative f ′ (ξ) of f in the direction ξ is defined by lim t→0 t −1 (f (γ ξ (t)) − f (p)), if the limit exists and is independent of the choice of a shortest path γ ξ .

3.2.
ε-open maps.Let f : X → Y be a continuous map between metric spaces and let ε be a (small) positive number.We say that f is ε-open if for any x ∈ X and any sufficiently small r > 0, we have B(f (x), εr) ⊂ f (B(x, r)).
We will use the following two lemmas from [13] regarding ε-open maps.The proofs are straightforward and do not rely on any curvature assumption.A map from an open subset of a GCBA space to Euclidean space is called differentiable if each coordinate function has directional derivative in any direction.Proof.We may use the 1-norm in R k .It suffices to show that for any p ∈ U and any v ∈ R k sufficiently close to f (p), there exists q ∈ U arbitrarily close to p such that |f (q)v| ≤ |f (p)v| − c(ε)|pq|.Then one can find r ∈ U such that f (r) = v and c(ε)|pr| ≤ |f (p)v| by a standard argument, which completes the proof.Here and below c(ε) denotes various positive constants such that c(ε) ≪ ε and κ(δ) denotes various positive function such that κ(δ) → 0 as δ → 0 (see Sec. 2).
We may assume v = 0.If f i (p) > 0 for some i, then the claim follows from the assumption (1).Similarly, if f i (p) < 0 for all i, then the claim follows from the assumption (2).Hence we may assume f i (p) ≤ 0 for all i and f i (p) = 0 for some i.For simplicity, we assume First, by the assumption (2), we choose .
By a standard argument using the assumption (1), one can find q 2 ∈ A 1 such that f 1 (q 2 ) = 0. Then we have

Next we consider
Using the assumption (1), one can find q ∈ A 2 such that f 2 (q) = 0.As above we obtain For i ≥ 3, we have We also have |pq| ≤ c(ε Then there exists q ∈ f −1 (f (p)) arbitrarily close to p such that q ′ is arbitrarily close to ξ.In particular, if {g i } i is a finite collection of differentiable, locally Lipschitz functions on U , then one can choose q so that g Proof.Choose a point q 1 ∈ U near p on a shortest path starting in the direction ξ so that |f (p)f (q 1 )| < δ|pq 1 |, where δ ≪ ε.Using the ε-openness of f , we can find q ∈ U such that f (q) = f (p) and ε|qq 1 | ≤ |f (p)f (q 1 )|.In particular |qq 1 | < κ(δ)|pq 1 | and hence ∠qpq 1 < κ(δ), which completes the proof of the first half.

Hurewicz fibrations.
We assume all maps are continuous.A map between topological spaces is called a Hurewicz fibration if it satisfies the homotopy lifting property with respect to any space.
The following two theorems from geometric topology, used by Lytchak-Nagano [9] for strainer maps, provide sufficient conditions for a map to be a Hurewicz fibration.Both are due to Ungar [16] and based on Michael's selection theorem.Definition 3.3.Let f : X → Y be a map between metric spaces.We say that f has locally uniformly contractible fibers if the following holds: for any x ∈ X and every neighborhood U of x, there exists a neighborhood V ⊂ U of x such that for any fiber Π of f intersecting V , the intersection Π ∩ V is contractible in Π ∩ U .

Infinitesimal properties
In this section we study the infinitesimal properties of noncritical maps and prove some lemmas which will be used in the next section.This section corresponds to [13, §2] (cf.[14, §2]) in CBB geometry.We use the notation ε, δ, c(ε), and κ(δ) introduced in Sec. 2. The choice of δ in this section will be determined by the proof of Proposition 4.5.
Throughout this section, Σ denotes a compact, geodesically complete CAT(1) space with diameter π.In case dim Σ = 0, we assume Σ is a set of finite points with pairwise distance π and not a singleton (for the sake of induction).Note that any two points in Σ with distance < π can be joined by a unique shortest path and that any shortest path in Σ can be extended to a shortest path of length π.As in the introduction, we define for ξ, η ∈ Σ.We call ξ ∈ Ant(ξ) an antipode of ξ (not necessarily unique).
We first show the following fact mentioned in the introduction.This section deals with the infinitesimal version of an (ε, δ)-noncritical map.
Next we prove (2) by induction on dim Σ.We may assume k ≥ 2 by Remark 4.3.For 1 ≤ j ≤ k, set We will show X j = ∅ inductively.
Next we show X j = ∅ by assuming X j−1 = ∅.Let x j be a closest point to ξ j in X j−1 .We show x j ∈ X j , i.e. |x j ξ j | = π/2.As before, {ξ ′ i } k i=2 is (c(ε), κ(δ))noncritical in Σ xj and η ′ is a regular direction.This holds in a small neighborhood of x j .Thus by Lemma 3.1, Remark 4.3, and the inductive hypothesis, the map for any i = j (≥ 2).Hence by Lemma 3.2, there exists x ∈ Σ near x j such that for any i = j (≥ 2).This contradicts the choice of where δ = c(ε) will be determined by the following argument.By Lemma 4.4(2), the same argument as above shows that there exists w ∈ Σ such that for any i ≥ 2, provided δ ≪ ε.This contradicts the choice of v.

Local properties
In this section we study the local properties of noncritical maps and prove Theorems 1.6 and 1.7.As before, we use the notation ε, δ, c(ε), and κ(δ).
Throughout this section, U denotes a tiny ball in a GCBA space X.Note that for any a, x, y ∈ U , we have ∠axy + ∠ayx ≤ π, where ∠ denotes the comparison angle.In particular if |ax| ≥ |ay|, we obtain ∠axy ≤ ∠axy ≤ π/2.We will often use this observation.
Let f = (|a 1 • |, . . ., |a k • |) be an (ε, δ)-noncritical map at p ∈ U in the sense of Definition 1.1 and let b ∈ U be as in the condition (2).In the terminology of the previous section, {a ′ i } k i=1 is an (ε, δ)-noncritical collection in Σ p with a regular direction b ′ .We also call b ′ a regular direction of f at p (note that this notation specifies not only the direction b ′ but also the point b).By the upper semicontinuity of angle and the local geodesic completeness, there exists a neighborhood V ⊂ U of p such that f is an (ε, δ)-noncritical map on V with a regular direction b ′ .We call V a regular neighborhood of f at p with respect to b.
If we want to estimate the size of a regular neighborhood, we use the following alternative definition.Let ρ be a positive number less than the radius of U .
In this case as well, we call b ′ a regular direction of f at p.
Remark 5.2.By the local geodesic completeness and the monotonicity of comparison angle, it suffices to consider the case x ∈ B(p, ρ) \ B(p, σ) in the above inequalities, where 0 < σ < ρ.
The above definition is equivalent to Definition 1.1 in the following sense: Proof.The "if" part follows from the local geodesic completeness and the monotonicity of angle.We show the "only if" part.By definition, we have |a ′ i ξ| + |a ′ j ξ| < 3π/2 + δ for any ξ ∈ Σ p .Then the first variation formula implies that ∠a i px + ∠a j px < 3π/2 + δ for any x sufficiently near p.Hence the first condition of Definition 5.1 holds for some ρ > 0. Similarly, the second condition holds.
The size of a regular neighborhood can be estimated as follows: Proof.By definition, we have ∠a i px + ∠a j px < 3π/2 + δ for any x ∈ B(p, ρ) \ {p}, where |a i p|, |a j p| > ρ.Hence if q ∈ B(p, ρδ) and x ∈ B(q, ρ/2) \ B(q, ρ/3), we have ∠a i qx + ∠a j qx < 3π/2 + κ(δ).By Remark 5.2, this implies the first condition of Definition 5.1 for q.Similarly, one can show the second condition.Now we will investigate the local properties of noncritical maps.First we prove Theorem 1.6.Now assume k = dim T p .We show f is injective near p.Suppose the contrary and let x, y ∈ V be distinct points sufficiently close to p such that f (x) = f (y).In particular we have ∠a i xy ≤ π/2 for any i as observed at the beginning of this section.We may assume |bx| ≥ |by|, which implies ∠bxy ≤ π/2 as well.Hence y ′ ∈ Σ x satisfies the assumption (1) of Lemma 4.4 for a noncritical collection {a ′ i } k i=1 and its regular direction b ′ (in particular we may assume k ≥ 2).Thus {(a ′ i ) ′ } k i=1 in Σ y ′ is a noncritical collection with a regular direction (b ′ ) ′ .Therefore by Proposition 4.5 (1) we have k ≤ dim Σ y ′ + 1 ≤ dim Σ x .On the other hand, if x is sufficiently close to p, we have dim T x ≤ k by the upper semicontinuity of local dimension.This is a contradiction.
From now on, we assume k < dim T p .We first observe that the fiber of f through p is not a singleton.
where k < dim T p .Then there exists a point of f −1 (f (p)) arbitrarily close to p.
Proof.Let b ′ be a regular direction of f at p and V a regular neighborhood.We may assume that there exist x j = y j ∈ V converging to p such that f (x j ) = f (y j ); otherwise f is a bi-Lipschitz open embedding near p, which contradicts k < dim T p .In particular we have ∠a i x j y j ≤ π/2 for any i as before.We may further assume |bx j | ≥ |by j | and hence ∠bx j y j ≤ π/2.
Extend the shortest path y j x j beyond x j to a shortest path y j z j of fixed length.The above estimates imply ∠a i x j z j ≥ π/2 and ∠bx j z j ≥ π/2.We may assume z j converges to z = p.By the upper semicontinuity of angle, we have ∠a i pz ≥ π/2 and ∠bpz ≥ π/2.For 0 ≤ j ≤ k, set .
We have now shown z ′ ∈ X 0 .Furthermore an inductive argument similar to the proof of Proposition 4.5 (2) shows X j = ∅ for any j ≥ 1.In particular for ξ ∈ X k we have f ′ (ξ) = 0. Thus the claim follows from Lemma 3.2.
Next we construct a local neighborhood retraction to the fiber of a noncritical map.Such a retraction for a strainer map (with much nicer properties) was constructed in [8, 9.1].Our construction is inspired by a similar argument in CBB geometry [5, 6.15].Here we use Definition 5.1 to specify the size of a regular neighborhood (Lemma 5.4).
Then, for any 0 < r < ρδ, there exists a continuous map Proof.Let b ′ be a regular direction of f at p. To simplify the notation, we set f i = |a i • | − |a i p| for any i.Let s = c(ε), which will be determined later as well as L. Define + as follows: for any x ∈ B(p, r), let R 1 (x) be the closest point to x on the intersection of the shortest path xb and Π s + .Let us check this definition works.By Lemmas 5.3 and 5.4, f is a (c(ε), κ(δ))-noncritical map with a regular direction b ′ on B(p, c(ε ) for any i by Remark 4.3.This means if one moves x toward b along the shortest path, the value of f i increases with velocity > c(ε).We show it reaches Π s + within time 2r/c(ε) (the case k = 1 is obvious).Let γ(t) be the shortest path xb with arclength parameter t.Then −r + c(ε)t ≤ f i (γ(t)) ≤ r + t.In particular if t = 2r/c(ε), we have Hence R 1 can be defined for s = (1 + 2c(ε) −1 ) −1 and L = 2c(ε) −1 + 1. Furthermore any y ∈ B(p, Lr) on the shortest path xb beyond R 1 (x) lies in the interior of Π s + (the case k = 1 is obvious).Indeed, where we take s smaller if necessary.Together with the uniqueness of shortest paths, this shows that R 1 is continuous.Clearly R 1 is the identity on Π s + .Next we construct a map R 2 : B(p, Lr) ∩ Π s + → Π − as follows: for any x ∈ B(p, Lr) ∩ Π s + , let R 2 (x) be the closest point to x on Π − .Since Π − is convex, triangle comparison shows that the closest point is uniquely determined.
On the other hand, since |xy| ≪ ρ we have ∠a i yā i > π − κ(δ).By Definition 5.1(1) we have ∠a j yā i ≤ ∠a j yā i < π/2 + κ(δ) for any j = i.This means that moving y toward āi increases the value of f j with velocity at most κ(δ).
Let y 1 be a point on the shortest path yā i sufficiently close to y.Then the assumption on y and the above observations yield The desired retraction R is now obtained as the composition of R 1 and R 2 .
Remark 5.7.One can also construct R 2 by using the gradient flows of semiconcave functions as in the CBB case [5, 6.15].The existence and uniqueness of such flows on GCBA spaces were shown in [7] (cf.[15]).We briefly recall the definition of the gradient (see [7] or [15] for more details).Let f be a semiconcave function defined on an open subset U of a GCBA space.Note that the directional derivative f ′ is well-defined in any direction and extended to the tangent cone by positive homogeneity.The gradient ∇ p f ∈ T p of f at p ∈ U is characterized by the following two properties: Here the absolute value denotes the distance from the vertex o of T p and the scalar product is defined by u, v := |u||v| cos ∠(u, v) for u, v ∈ T p .More specifically, if max f ′ | Σp > 0 then ∇ p f = f ′ (ξ max )ξ max , where ξ max ∈ Σ p is the unique maximum point of f ′ | Σp ; otherwise ∇ p f = o.The gradient flow of f is the flow along the gradient of f .The alternative construction of R 2 is as follows.Set a concave function F := min{−f i , 0} and consider the gradient flow Φ t of F on B(p, c(ε) −1 r), where t ≥ 0. We will define R 2 := Φ T for some sufficiently large T > 0.
Clearly Φ t fixes Π − .Suppose x / ∈ Π − .Let I(x) be the set of indices 1 ≤ i ≤ k such that F (x) = −f i (x).Then it follows from the noncriticality of f that , where the first inequality follows from |a ′ i ā′ i | = π, the second one follows from the property of the gradient, and the last one follows from the noncriticality of f .We show that Φ t pushes B(p, Lr) ∩ Π s + to Π inside Π s + within time T = Lr/c(ε).Let x ∈ B(p, Lr) ∩ Π s + .It suffices to show that y = Φ t (x) lies in Π s + for sufficiently small t > 0. We may assume x is on the boundary of Π s + , that is, f i (x) = sf j (x) for some i = j.This implies i / ∈ I(x) and j ∈ I(x).By the observation above, we have Since any metric ball contained in a tiny ball is contractible, we have: where L = c(ε) −1 .In particular, f has locally uniformly contractible fiber on B(p, r) (see Definition 3.3).
The following theorem is a direct consequence of Theorem 3.4, Theorem 1.6, and Corollary 5.8.Theorem 5.9.Let f : U → R k be a distance map defined on a tiny ball U that is noncritical on an open subset To prove Theorem 1.7, we need to control the boundary of a ball in the fiber.
Proof.Let x ∈ ∂B(p, r) ∩ f −1 (f (p)).In particular ∠a i px ≤ π/2.Let p′ be an arbitrary antipode of p ′ in Σ x .By the upper semicontinuity of angle and the local geodesic completeness, we have |a It remains to show that there exists η ∈ Σ x such that We may assume that f is (ε, δ)-noncritical on B(p, r).In particular there exists ξ ∈ Σ x such that |a ′ i ξ| < π/2 − ε.We move ξ toward p′ to get η as above.Fix an antipode p′ of p ′ in Σ x .Recall that p is a (1, δ)-strainer at x if r is small enough (see [8, 7.3]).In particular |p ′ p′ | < 2δ (see [8, 6.3]).Let ā′ i be an arbitrary antipode of a Finally we prove Corollary 1.9.The proof is an easy application of the second half of Theorem 1.6.Strictly speaking we need its global version for CAT(0) spaces, which easily can be verified.Remark 5.12.Under the assumption of Corollary 1.9, the diameter of Σ (with respect to the original metric) is less than 2π; in particular the original metric is uniformly bi-Lipschitz equivalent to the π-truncation.This is observed as follows.Let {ξ i } n+1 i=1 and η be as in the assumption.Then Lemma 4.4(2) and Proposition 4.5 (1) imply that they are π/2-dense in Σ, whereas Remark 4.3 shows that they are π-close to each other.
Proof of Corollary 1.9.Consider the Euclidean cone K over Σ and let γ i be the ray starting at the vertex o in the direction ξ i .Let f i : K → R be the Busemann function with respect to γ i : where the absolute value denotes the distance from o. Set f := (f 1 , . . ., f n+1 ) : K → R n+1 .Observe that f is a (normalized) limit of (ε, δ)-noncritical maps to R n+1 defined on arbitrarily large neighborhoods of o, which are c(ε)-open embeddings by the global version of Theorem 1.6 for CAT(0) spaces.Therefore f is a bi-Lipschitz homeomorphism.Identify Σ with the unit sphere in K centered at o and S n with the unit sphere in R n+1 centered at 0. We may use the extrinsic metrics of Σ and S n , that is, the restrictions of the metrics of K and R n+1 , respectively.Define f : Σ → S n by f (x) := f (x)/|f (x)|.
Let us prove that f is a bi-Lipschitz homeomorphism.Note that f maps a ray emanating from o to a ray emanating from 0. Since f is surjective, this implies that f is also surjective.The Lipschitz continuity of f follows from that of f by the following calculation: We define the antipodal distance | , | on Σ by |ξη| := sup x∈Σ |ξx| + |ηx| − π for ξ, η ∈ Σ, where | , | denotes the distance on Σ.The antipodal distance is not a distance in the usual sense.The reason for this name is the following equivalent definition.Let Ant(ξ) denote the set of all antipodes ξ of ξ, i.e. |ξ ξ| = π.Then it easily follows from the geodesic completeness that |ξη| = sup ξ∈Ant(ξ) | ξη| = sup η∈Ant(η) |ξ η| (see Lemma 4.1 for the detail).Thus the antipodal distance on spaces of directions reflects the branching phenomena of shortest paths in GCBA spaces.
Then f is c(ε)-open on U with respect to the Euclidean norm in R k .The choices of δ and c(ε) depend only on k and ε.

Theorem 3 . 4 (
[16, Theorem 1]).Let X, Y be finite-dimensional, compact metric spaces and let Y be an ANR.Let f : X → Y be an open, surjective map with locally uniformly contractible fibers.Then f is a Hurewicz fibration.Theorem 3.5 ([16, Theorem 2]).Let X, Y be finite-dimensional, locally compact metric spaces.Let f : X → Y be an open, surjective map with locally uniformly contractible fibers.Assume all fibers of f are contractible.Then f is a Hurewicz fibration.
which means y ∈ Π s + .Hence we can define R 2 := Φ T on B(p, Lr) ∩ Π s + (the image of R 2 is contained in B(p, Lr + Lr/c(ε)), so we need to replace the constant L with L + L/c(ε) in the statement of Proposition 5.6).