On the definition of Alexandrov space
aa r X i v : . [ m a t h . DG ] F e b On the definition of Alexandrov space Abstract.
This paper shows that, in the definition of Alexandrov space with lower ([BGP]) or upper([AKP]) curvature bound, the original conditions can be replaced with much weaker ones, which canbe viewed as comparison versions of the second variation formula in Riemannian geometry (and thus ifwe define Alexandrov spaces using these weakened conditions, then the original definition will becomea local version of Toponogov’s Comparison Theorem on such spaces). As an application, we give anew proof for the Doubling Theorem by Perel’man.
Key words.
Alexandrov space, variation formula, curvature bound, Doubling Theorem.
Mathematics Subject Classification (2020) : 53C20, 51F99.
Alexandrov spaces with lower curvature bound have been developed systematically by Burago-Gromov-Perel’man in [BGP]. Afterwards, Alexandrov geometry has been studied intensively by many ge-ometrists. To define an intrinsic metric space to be an Alexandrov space with lower curvature bound,several equivalent conditions are given in [BGP].As in [BGP], for convenience of formulations, we always assume that an intrinsic metric space X satisfies that around any x ∈ X there is a neighborhood U x such that any two distinct points in U x can be joined by a minimal geodesic (i.e. shortest path). We will denote respectively by [ xy ] and | xy | a minimal geodesic and the distance between x and y in X . Moreover, we denote by S k the completeand simply connected 2-dimensional space form with constant curvature k .We now show a definition of an Alexandrov space with lower curvature bound in [BGP]. Definition 0.1.
An intrinsic metric space X is called an Alexandrov space with curvature > k ifaround any x ∈ X there is a neighborhood U x such that the following condition is satisfied:(1) To any p ∈ U x and any [ qr ] ⊂ U x , we associate ˜ p ∈ S k and [˜ q ˜ r ] ⊂ S k with | ˜ p ˜ q | = | pq | , | ˜ p ˜ r | = | pr | and | ˜ q ˜ r | = | qr | . Then, for ALL s ∈ [ qr ] and ˜ s ∈ [˜ q ˜ r ] with | qs | = | ˜ q ˜ s | , | ps | > | ˜ p ˜ s | . One of our main results shows that condition (1) in Definition 0.1 can be much weaker.
Theorem A.
Let X be an intrinsic metric space. Then X is an Alexandrov space with curvature > k if around any x ∈ X there is a neighborhood U x such that the following is satisfied: (A) To any p ∈ U x and any [ qr ] ⊂ U x , we associate ˜ p ∈ S k and [˜ q ˜ r ] ⊂ S k with | ˜ p ˜ q | = | pq | , | ˜ p ˜ r | = | pr | and | ˜ q ˜ r | = | qr | . Then, for s ∈ [ qr ] and ˜ s ∈ [˜ q ˜ r ] with | qs | = | ˜ q ˜ s | , lim sup s → q | ps | − | ˜ p ˜ s || qs | > . (0 . | ps | > | ˜ p ˜ s | + o ( | qs | ) for s sufficiently close to q .This is clearly related to the geodesic variation on Riemannian manifolds. It is well known that the Supported by NSFC 11971057 and BNSF Z190003. The corresponding author (E-mail: [email protected]). p, q, r ∈ X and ˜ p, ˜ q, ˜ r ∈ S k with | ˜ p ˜ q | = | pq | , | ˜ p ˜ r | = | pr | and | ˜ q ˜ r | = | qr | , we denote by ˜ ∠ k pqr the angle ∠ ˜ p ˜ q ˜ r on S k . Theorem B.
Let X be an intrinsic metric space. Then X is an Alexandrov space with curvature > k if around any x ∈ X there is a neighborhood U x such that the following is satisfied: (B) For any small ǫ > , there is δ > such that any p ∈ U x and any [ qr ] ⊂ U x satisfy | ps | | ¯ p ¯ s | + o ( | qs | ) with o ( | qs | ) ǫ | qs | , for all s ∈ [ qr ] with | qs | δ, (0 . where ¯ p and ¯ s belong to a triangle △ ¯ p ¯ q ¯ s ⊂ S k with | ¯ p ¯ q | = | pq | , | ¯ q ¯ s | = | qs | and ∠ ¯ p ¯ q ¯ s = lim sup t → q, t ∈ [ qr ] ˜ ∠ k pqt . Here, the infinitesimal o ( | qs | ) (as s → q ) has to satisfy the requirement of the uniformity, i.e. o ( | qs | ) ǫ | qs | when | qs | δ . As a counterexample, one can consider the space of three rays startingfrom a common point (which is not an Alexandrov space with lower curvature bound).In Section 2, we will provide an equivalent version and a bit stronger version of condition (B)(see (B) ′ and (B) ′′ ), and will show the relations in conditions (A), (B), and other equivalent versionsof condition (1) in [BGP]. It turns out that, among these conditions, condition (B) should be theweakest one to define an Alexandrov space with lower curvature bound (see Subsections 1.2 and 1.4). Remark 0.2.
Conditions (A) and (B) have their corresponding ‘local’ versions, under which TheoremsA and B are still true. Precisely, for example, the ‘local’ version of (B) is:
For any p ∈ U x , around any q ∈ U x there is a neighborhood U q ,p such that for any small ǫ > there is δ > such that (0.2) is satisfied for any [ qr ] ⊂ U q ,p . Remark 0.3.
It is well known that, in proving Toponogov’s Comparison Theorem on Riemannianmanifolds, the fundamental tool is the second variation formula. Thereby, if we define Alexandrovspaces with lower curvature bound using condition (A) or (B), then the definition in [BGP] can beviewed as a local version of Toponogov’s Comparison Theorem on such spaces.
Remark 0.4.
Since conditions (A) and (B) are weaker than condition (1) (and other equivalentconditions in [BGP]), it will be possibly easier to judge an intrinsic metric space to be an Alexandrovone by verifying condition (A) or (B) on the space. For instance, we can re-prove the DoublingTheorem by Perel’man in this way (see Section 3).
Remark 0.5.
An intrinsic metric space X is called an Alexandrov space with curvature k if‘ | ps | > | ˜ p ˜ s | ’ −→ ‘ | ps | | ˜ p ˜ s | ’ in condition (1) ([AKP]).Consequently and similarly, X will be an Alexandrov space with curvature k if we make the followingchanges in Theorems A and B (similarly in other conditions in Section 1):‘lim sup’ −→ ‘lim inf’, ‘ ǫ ’ −→ ‘ − ǫ ’ in (0.2), ‘ > ’ ←→ ‘ ’ except the ‘ ’ of ‘ | qs | δ ’ in (0.2).In order to prove this, we just need to make the following changes in the corresponding arguments inSections 1 and 2:‘lim sup’ ←→ ‘lim inf’, ‘+ ǫ ’ −→ ‘ − ǫ ’, ‘ > ’ ←→ ‘ ’, ‘max’ ←→ ‘min’. Here, we in fact need an assumption that each of | pq | , | pr | , | qr | is less than π √ k if k >
2n the rest of the paper, we mainly give a proof of Theorem B in Section 2, while Theorem Bimplies Theorem A by the relation between conditions (A) and (B) in Section 1 (see Remark 1.1). Asan application, we will give a new proof for the Doubling Theorem in Section 3. And in Appendix,we will make a detailed study on ‘ f ′′ ( t ) + kf ( t ) ( > ) 0 in the support sense’. In the present paper, we will use the following model functions (cf. [Pet]):sn k ( ρ ) , √ k sin( √ kρ ) , k > ρ, k = 0 √− k sinh( √− kρ ) , k < , ct k ( ρ ) , sn ′ k ( ρ )sn k ( ρ ) , f k ( ρ ) , k Ä − cos( √ kρ ) ä , k > ρ , k = 0 − k (cid:0) cosh( √− kρ ) − (cid:1) , k < . Condition (B) can be formulated as follows:(B) ′ For any small ǫ > , there is δ > such that any p ∈ U x and any [ qr ] ⊂ U x satisfy − cos ˜ ∠ k pqs − cos ∢ pqr + o ( | qs | ) with o ( | qs | ) ǫ , for all s ∈ [ qr ] with | qs | δ, (1 . where ∢ pqr , lim sup t → q, t ∈ [ qr ] ˜ ∠ k pqt . In fact, by the Law of Cosine on S k , for s and ¯ s in (0.2) we have that | ps | = | pq | − cos ˜ ∠ k pqs · | qs | + 12 ct k ( | pq | ) sin ˜ ∠ k pqs · | qs | + o ( | qs | ) , | ¯ p ¯ s | = | pq | − cos ∢ pqr · | qs | + 12 ct k ( | pq | ) sin ∢ pqr · | qs | + o ( | qs | ) , (1 . o i ( | qs | ) is a higher order infinitesimal of | qs | as | qs | →
0; or equivalently, f k ( | ps | ) = f k ( | pq | ) − sn k ( | pq | ) cos ˜ ∠ k pqs · | qs | + 12 (1 − kf k ( | pq | )) · | qs | + o ( | qs | ) ,f k ( | ¯ p ¯ s | ) = f k ( | pq | ) − sn k ( | pq | ) cos ∢ pqr · | qs | + 12 (1 − kf k ( | pq | )) · | qs | + o ( | qs | ) . (1 . ′ . Let p, q, r, s and ˜ p, ˜ q, ˜ r, ˜ s be the notations in condition (A). By the Law of Cosine on S k , for s ∈ [ qr ]sufficiently close to q and ˜ s ∈ [˜ q ˜ r ] with | qs | = | ˜ q ˜ s | , we have that | ps | = | pq | − cos ˜ ∠ k pqs · | qs | + o ( | qs | ) , | ˜ p ˜ s | = | pq | − cos ˜ ∠ k pqr · | qs | + o ( | qs | ) . It then is easy to see that (0.1) in condition (A) is equivalent to that − cos ˜ ∠ k pqr − lim inf s → q cos ˜ ∠ k pqs .Thereby, by replacing [ qr ] with [ qs ] for any s = q in condition (A), we have that − cos ˜ ∠ k pqs − cos Ç lim sup t → q, t ∈ [ qr ] ˜ ∠ k pqt å , or equivalently , ˜ ∠ k pqs lim sup t → q, t ∈ [ qr ] ˜ ∠ k pqt, (1 . ′ , so of (B).3 emark 1.1. Let U x satisfy condition (A), and let p ∈ U x and [ r r ] ⊂ U x . Since condition (A) is aspecial case of (B), for any q ∈ [ r r ] ◦ (the interior part of [ r r ]) we have that ∢ pqr + ∢ pqr π where ∢ pqr i , lim sup t → q, t ∈ [ qr i ] ˜ ∠ k pqt (see Lemma 2.1 below), which together with (1.4) implies that˜ ∠ k pqr + ˜ ∠ k pqr π. Then, without involving Lemma 2.2 below, this implies that U x satisfies condition (1) in Definition 0.1by Alexandrov’s lemma (Lemma 2.5 in [BGP], cf. [AKP]). If it is under the local version of condition(A) (see Remark 0.2), one just only need to repeat Alexandrov’s lemma finite times to see that U x satisfies condition (1) in Definition 0.1. It is obvious that condition (1) in Definition 0.1 has an equivalent version as follows ([BGP]):(2) For any p ∈ U x and any [ qr ] ⊂ U x , ˜ ∠ k pqs with s ∈ [ qr ] is non-increasing with respect to | qs | .Apparently, condition (2) implies that lim t → q, t ∈ [ qr ] ˜ ∠ k pqt exists and˜ ∠ k pqs lim t → q, t ∈ [ qr ] ˜ ∠ k pqt for ALL s ∈ [ qr ] \ { q } . (1 . pq ] and [ qr ] in an Alexandrov space with curvature > k , we candefine an angle between them at q to be ∠ pqr , lim x, y → q ˜ ∠ k xqy, where x ∈ [ pq ] and y ∈ [ qr ] ([BGP]).And thus, given [¯ p ¯ q ] , [¯ q ¯ r ] ⊂ S k with | ¯ p ¯ q | = | pq | , | ¯ q ¯ r | = | qr | and ∠ ¯ p ¯ q ¯ r = ∠ pqr , we have that | ps | | ¯ p ¯ s | for all s ∈ [ qr ] and ¯ s ∈ [¯ q ¯ r ] with | ¯ s ¯ q | = | sq | ([BGP]). Moreover, if q is an interior point of some [ rr ′ ] inaddition, then ∠ pqr + ∠ pqr ′ = π . Conversely, these properties together can be viewed as a sufficientcondition for curvature > k . Namely, condition (1) in Definition 0.1 can be replaced by the followingconditions (cf. [BGP]):(3-1) For any [ pq ] and [ qr ] ⊂ U x , ∠ pqr can be defined so that ∠ pqr + ∠ pqr ′ π if q is an interiorpoint of some [ rr ′ ].(3-2) To any [ pq ] and [ qr ] ⊂ U x , we associate [¯ p ¯ q ] , [¯ q ¯ r ] ⊂ S k with | ¯ p ¯ q | = | pq | , | ¯ q ¯ r | = | qr | and ∠ ¯ p ¯ q ¯ r = ∠ pqr . Then | ps | | ¯ p ¯ s | for ALL s ∈ [ qr ] and ¯ s ∈ [¯ q ¯ r ] with | ¯ s ¯ q | = | sq | .Here, condition (3-1) cannot be cancelled (hint: one can define angles on the space of three raysstarting from a common point so that (3-2) is satisfied, but (3-1) not). Inspired by (1.1) and (1.5), we can present the following condition, a bit stronger version of condition(B) ′ (so of (B)) as well as a much weaker version of condition (2):(B) ′′ For any small ǫ > , there is δ > such that any p ∈ U x and any [ qr ] ⊂ U x satisfy ˜ ∠ k pqs lim sup t → q, t ∈ [ qr ] ˜ ∠ k pqt + o ( | qs | ) with o ( | qs | ) ǫ , for all s ∈ [ qr ] with | qs | δ. (1 . To any [ pq ] and [ qr ] ⊂ U x , we associate [¯ p ¯ q ] , [¯ q ¯ r ] ⊂ S k with | ¯ p ¯ q | = | pq | , | ¯ q ¯ r | = | qr | and ∠ ¯ p ¯ q ¯ r = ∠ pqr . Then for s ∈ [ qr ] sufficiently close to q and ¯ s ∈ [¯ q ¯ r ] with | ¯ q ¯ s | = | qs | , | ps | | ¯ p ¯ s | + o ( | qs | ) . (1 . Remark 1.2.
In proving Theorem B, we will first show that condition (3-1) is satisfied throughLemma 2.1. This plus (1.7), partial information of (0.2), guarantees Lemma 2.2. Then Theorem Bfollows almost immediately. Namely, from the proof of Theorem B, we can conclude that an intrinsicmetric space is an Alexandrov space with curvature > k if it satisfies conditions (3-1) and (C). In our proof, we sometimes substitute condition (B) ′ for (B) because they are equivalent to each other(see Subsection 1.1). First of all, we list two lemmas, and then prove Theorem B by assuming them. Lemma 2.1 (Key Lemma) . Let U x be the neighborhood of x ∈ X satisfying condition (B) ′ , and let γ ( t ) | [0 ,µ ] be an arc-length parameterized minimal geodesic in U x . Then for any p ∈ U x and t ∈ (0 , µ ) , lim sup t → t +0 ˜ ∠ k pγ ( t ) γ ( t ) + lim sup t → t − ˜ ∠ k pγ ( t ) γ ( t ) π. (2 . Lemma 2.2.
Let U x be the neighborhood of x ∈ X satisfying condition (B) , and let γ ( t ) | [0 ,µ ] be anarc-length parameterized minimal geodesic in U x . Then for any p ∈ U x , g ( t ) , f k ( | pγ ( t ) | ) satisfies g ′′ ( t ) + kg ( t ) in the support sense , ∀ t ∈ (0 , µ ) . (2 . g ′′ ( t ) bounded from above in the support sense, please referto Appendix. In proving Lemma 2.2, Lemma 2.1 will play a crucial role. Proof of Theorem B.
Let U x be the neighborhood of x ∈ X satisfying condition (B), and let γ ( t ) | [0 ,µ ] be an arc-length parameterized minimal geodesic in U x . To γ ( t ) | [0 ,µ ] and any p ∈ U x , we associate an arc-length parameterized minimal geodesic ˜ γ ( t ) | [0 ,µ ] and a point ˜ p in S k with | ˜ p ˜ γ (0) | = | pγ (0) | and | ˜ p ˜ γ ( µ ) | = | pγ ( µ ) | . According to condition (1) in Definition 0.1, it suffices to show that | pγ ( t ) | > | ˜ p ˜ γ ( t ) | ∀ t ∈ [0 , µ ] . (2 . g ( t ) , f k ( | pγ ( t ) | ) and ˜ g ( t ) , f k ( | ˜ p ˜ γ ( t ) | ). Note that g (0) = ˜ g (0) and g ( µ ) = ˜ g ( µ ). Moreover, itis well known that ([PP], cf. [Pet]) ˜ g ′′ ( t ) + k ˜ g ( t ) = 1 ∀ t ∈ (0 , µ ) , then by Lemma 2.2 it is clear that, in the support sense,( g ( t ) − ˜ g ( t )) ′′ + k ( g ( t ) − ˜ g ( t )) ∀ t ∈ (0 , µ ) . (2 . g ( t ) − ˜ g ( t ) > t ∈ [0 , µ ],i.e. f k ( | pγ ( t ) | ) > f k ( | ˜ p ˜ γ ( t ) | ), which is equivalent to (2.3). (Here, it should be needed that µ < π √ k if k >
0, while this can be guaranteed indeed because U x can be selected sufficiently small.) Remark 2.3.
In fact, [PP] has shown that an intrinsic metric space X is an Alexandrov spacewith curvature > k if and only if (2.2) is satisfied for any p ∈ X and any minimal geodesic γ ( t ) ⊂ X , but the direct relation between (2.2) and (2.3) (i.e. condition (0.1)) has not been pointed out(cf. Proposition 1.6 in [PP]). Moreover, on Riemannian manifolds, [Pet] has proven Toponogov’sComparison Theorem just based on the inequality in (2.4). These prompt us to see that (2.4) with g (0) − ˜ g (0) = g ( µ ) − ˜ g ( µ ) = 0 implies g ( t ) − ˜ g ( t ) >
0, and give a detailed proof for it in Appendix.5ext, we verify Lemma 2.1, the key lemma.
Proof of Lemma 2.1.
As in (B) ′ , we let ∢ pγ ( t ) γ ( µ ) , lim sup t → t +0 ˜ ∠ k pγ ( t ) γ ( t ) and ∢ pγ ( t ) γ (0) , lim sup t → t − ˜ ∠ k pγ ( t ) γ ( t ). Claim : ∢ pγ ( t ) γ ( µ ) lim inf t → t − ∢ pγ ( t ) γ ( µ ).According to (1.1) in condition (B) ′ , for any small ǫ > there is δ > − cos ˜ ∠ k pγ ( t ) γ ( t + τ ) − cos ∢ pγ ( t ) γ ( µ ) + ǫ for all t ∈ (0 , µ ) and any τ ∈ (0 , δ ) with t + τ µ . And, by the continuity of | pγ ( t ) | , it is clear thatlim t → t − ˜ ∠ k pγ ( t ) γ ( t + τ ) = ˜ ∠ k pγ ( t ) γ ( t + τ ) . It then follows that − cos ˜ ∠ k pγ ( t ) γ ( t + τ ) − cos Ç lim inf t → t − ∢ pγ ( t ) γ ( µ ) å + ǫ. Note that the claim follows as long as we let τ → ǫ → f ( t ) , | pγ ( t ) | , and for any t ∈ (0 , µ ) we set f ′− , min ( t ) , lim inf t → t − f ( t ) − f ( t ) t − t and f ′ + , max ( t ) , lim sup t → t +0 f ( t ) − f ( t ) t − t . Observe that f ′− , min ( t ) = cos ∢ pγ ( t ) γ (0) and f ′ + , max ( t ) = − cos ∢ pγ ( t ) γ ( µ ) . In fact, as t → t − , it is easy to see that f ( t ) − f ( t ) = − cos ˜ ∠ k pγ ( t ) γ ( t ) · ( t − t ) + o ( t − t )(via the Law of Cosine on S k ). And thus f ′− , min ( t ) = lim inf t → t − Ä cos ˜ ∠ k pγ ( t ) γ ( t ) ä = cos lim sup t → t − ˜ ∠ k pγ ( t ) γ ( t ) ! = cos ∢ pγ ( t ) γ (0) . Similarly, we have that f ′ + , max ( t ) = − cos ∢ pγ ( t ) γ ( µ ).Another important observation is that lim inf t → t − f ′ + , max ( t ) f ′− , min ( t ) (by the Sublemma below).As a result, we have thatcos ∢ pγ ( t ) γ (0) > lim inf t → t − ( − cos ∢ pγ ( t ) γ ( µ )) > − cos Ç lim inf t → t − ∢ pγ ( t ) γ ( µ ) å , which together with the claim above implies thatcos ∢ pγ ( t ) γ (0) > − cos ∢ pγ ( t ) γ ( µ ) , or equivalently, ∢ pγ ( t ) γ (0) + ∢ pγ ( t ) γ ( µ ) π. Namely, (2.1) has been verified, and thus the proof of the lemma is finished.6 ublemma.
Let f : ( a, b ) → R be a continuous function. Then for any t ∈ ( a, b ) , lim inf t → t − f ′ + , max ( t ) f ′− , min ( t ) . Proof . (This might be a known result in real analysis.) We first note that, for any [ c, d ] ⊂ ( a, b ), thereexists x ∈ [ c, d ) such that f ′ + , max ( x ) , lim sup t → x + f ( t ) − f ( x ) t − x f ( d ) − f ( c ) d − c (it can be seen similarly as Differential Mean Value Theorem, cf. [MV]). Meantime, by definition, f ′− , min ( t ) = lim inf t → t − f ( t ) − f ( t ) t − t . Therefore, there exists x t ∈ [ t, t ) (here a < t < t ) such thatlim inf t → t − f ′ + , max ( t ) lim inf t → t − f ′ + , max ( x t ) f ′− , min ( t ) . (cid:3) Remark 2.4.
Note that in proving Lemma 2.1, we use only the uniformity in (1.1), i.e. “ − cos ˜ ∠ k pqs − cos ∢ pqr + ǫ for all s ∈ [ qr ] with | qs | δ ”, without involving o ( | qs | ).Finally, in order to complete the whole proof of Theorem B, we just need to verify Lemma 2.2. Proof of Lemma 2.2.
Let U x satisfy condition (B) (or (B) ′ ), and let p ∈ U x and [ qr ] ⊂ U x , and let s, ¯ p, ¯ s be thenotations in (0.2). According to (0.2), it holds that | ps | | ¯ p ¯ s | + o ( | qs | ), which is equivalent to f k ( | ps | ) f k ( | ¯ p ¯ s | ) + o ( | qs | ). Then by the second equality of (1.3), we have that f k ( | ps | ) f k ( | pq | ) − sn k ( | pq | ) cos ∢ pqr · | qs | + 12 (1 − kf k ( | pq | )) · | qs | + o ( | qs | ) , (2 . ∢ pqr , lim sup t → q, t ∈ [ qr ] ˜ ∠ k pqt . We now consider the γ ( t ) | [0 ,µ ] ⊂ U x in the lemma, and let t ∈ (0 , µ ).By Lemma 2.1, it holds that ∢ pγ ( t ) γ (0) + ∢ pγ ( t ) γ ( µ ) π . Then, via (2.5) it is easy to see that f k ( | pγ ( t + τ ) | ) f k ( | pγ ( t ) | ) − sn k ( | pγ ( t ) | ) cos ∢ pγ ( t ) γ ( µ ) · τ + 12 (1 − kf k ( | pγ ( t ) | )) · τ + o ( τ )(where | τ | is sufficiently small). Then by Definition A.1, it is clear that g ′′ ( t ) − kg ( t ) in thesupport sense (where g ( t ) = f k ( | pγ ( t ) | )); i.e., the proof of the lemma is completed. Remark 2.5.
When applying Theorem B to judge an intrinsic metric space X to be an Alexandrovone, we can in fact verify a bit weaker version of condition (B): For any p ∈ U x and [ r r ] ⊂ U x , (0.2) holds with respect to [ qr i ] for all but at most a finite number of q ∈ [ r r ] ◦ , and (0.2) is replacedby “ | ps | | ¯ p ¯ s | + ǫ | qs | for all s ∈ [ qr i ] with | qs | δ ” at each exceptional q . Alternatively, inspiredby Remark 1.2, we can verify that: (3-1) holds on U x ; and for any p ∈ U x and [ r r ] ⊂ U x , (1.7) incondition (C) holds with respect to [ qr i ] for all but at most a finite number of q ∈ [ r r ] ◦ , and at eachexceptional q it holds that | ps | | pq | − cos ∠ pqr i · | qs | + o ( | qs | ) for s ∈ [ qr i ] sufficiently close to q . Thisguarantees that f + , max ( t ) f − , min ( t ) if we set f ( t ) , | pγ ( t ) | , where γ ( t ) | [0 , | r r | ] denotes [ r r ] with | r γ ( t ) | = t . Then we can apply Remark A.4 instead of Proposition A.2 to see that X is of curvature > k (cf. the end of the proof of Theorem B). 7 A new proof of the Doubling Theorem
Let’s first state the Doubling Theorem by Perel’man ([Pe]).
Theorem 3.1 . Let X be a complete n -dimensional Alexandrov space with curvature > k and withnonempty boundary. Then its doubling D ( X ) with canonical metric is a complete n -dimensionalAlexandrov space with curvature > k and with empty boundary. We can view D ( X ) as X ∪ ∂X ¯ X , where ¯ X is a copy of X . Between any x and y in D ( X ), thedistance | xy | is just the distance between them in X (or ¯ X ) if both x and y lie in X (or ¯ X ), otherwise, | xy | , min z ∈ ∂X {| xz | + | zy |} ([Pe]). Then it is easy to see the following property of D ( X ). Fact 3.2 (cf. [Pe]).
Any [ xy ] ⊂ D ( X ) satisfies that [ xy ] ◦ passes ∂X at most once or [ xy ] ⊂ ∂X ( ⊂ D ( X ) ), and [ xy ] ◦ passes ∂X once if and only if x ∈ X ◦ and y ∈ ¯ X ◦ or vice versa. In the proof of Theorem 3.1, we will use the following fundamental property of X . Fact 3.3 (cf. [BGP]).
For q ∈ ∂X and small ε > , max ζ ∈ ∂ Σ q X min {| ζ ↑ uq | : u ∈ ∂X, | qu | = ε } → as ε → , where Σ q X is the space of directions of X at q and ↑ uq ∈ Σ q X is the direction of some [ qu ] .Proof of Theorem 3.1. Not that we can apply induction on the dimension n starting with the trivial case where n = 1(note that X is an interval of length π by convention, cf. [BGP]).We first show that D ( X ) satisfies condition (3-1) in Subsection 1.3 globally; i.e., for any [ pq ] and[ qr ] ⊂ D ( X ), ∠ pqr can be defined so that ∠ pqr + ∠ pqr ′ π if q is an interior point of some [ rr ′ ]. Byinduction, for any x ∈ ∂X , D (Σ x X ) = Σ x X ∪ ∂ Σ x X Σ x ¯ X is a complete ( n − > ∠ pqr tobe the distance between the directions of [ pq ] and [ qr ] at q in D (Σ q X ) or Σ q X according to q ∈ ∂X or not respectively. And if q ∈ [ rr ′ ] ◦ , there are only the following three possible cases. • q ∈ X ◦ or ¯ X ◦ : It is clear that ∠ pqr + ∠ pqr ′ = π by (3-1) on X (which is of curvature > k ). • [ rr ′ ] ⊂ ∂X : It also holds that ∠ pqr + ∠ pqr ′ = π by (3-1) on X or ¯ X for p ∈ X or ¯ X respectively. • q ∈ ∂X , and r ∈ X ◦ and r ′ ∈ ¯ X ◦ (or vice versa): In this case, any [ qr ] lies in X and [ qr ′ ] liesin ¯ X . By the first variation formula on X and ¯ X , it has to hold that | ↑ rq ↑ r ′ q | = π in D (Σ q X ). As aresult, ∠ pqr + ∠ pqr ′ = π , and there is a unique minimal geodesic between q and r (and r ′ ).Next, we let [ pq ] and [ qr ] be two minimal geodesics in D ( X ) with p ∈ ¯ X with | pq | + | qr | < π √ k if k >
0. And let △ ¯ p ¯ q ¯ r be a triangle in S k with | ¯ p ¯ q | = | pq | , | ¯ q ¯ r | = | qr | and ∠ ¯ p ¯ q ¯ r = ∠ pqr , and let s ∈ [ qr ]and ¯ s ∈ [¯ q ¯ r ] satisfy | ¯ s ¯ q | = | sq | . Claim 1 . If q ∈ ∂X , then for s ∈ [ qr ] sufficiently close to q we have that | ps | | ¯ p ¯ s | + o ( | qs | ) . (3 . Claim 2 . If q ∈ X ◦ or [ qr ] ⊂ ¯ X , then for s ∈ [ qr ] sufficiently close to q we have that | ps | | ¯ p ¯ s | + o ( | qs | ) . (3 . D ( X ) also satisfies the weakerversion of condition (C) in Remark 2.5 almost globally. Hence, according to Remark 2.5, we canconclude that D ( X ) is of curvature > k , and then it is not hard to see that Σ x D ( X ) = D (Σ x X ) It also holds that max ζ ∈ Σ q X min {| ζ ↑ uq | : u ∈ X, | qu | = ε } → ε → q ∈ ∂X or not. An essential reasonfor this is that ( ε X, q ) → T q with ( ε ∂X, q ) → ∂T q as ε →
0, where T q is the tangent cone of X at q ([BGP]).
8r Σ x X according to x ∈ ∂X or not respectively, which implies that D ( X ) has no boundary point.Namely, the Doubling Theorem follows (note that the completeness of D ( X ) is obvious).We now verify Claim 1. Note that (3.1) is guaranteed by the first variation formula on X or ¯ X if p ∈ ∂X or [ qr ] ⊂ ¯ X . I.e., we can assume that p ∈ ¯ X ◦ and so ↑ pq ∈ (Σ q ¯ X ) ◦ , and ↑ rq ∈ (Σ q X ) ◦ . Thenthere is ξ ∈ ∂ Σ q X such that | ↑ pq ξ | + | ξ ↑ rq | = | ↑ pq ↑ rq | (= ∠ pqr ) in D (Σ q X ). Here, the difficulty is thatthere might be no [ qu ] ⊂ ∂X with ↑ uq = ξ . Nevertheless, according to Fact 3.3, for sufficiently small ǫ > ε ǫ such that for any positive t ε ǫ there is u ∈ ∂X with | qu | = t and | ↑ uq ξ | ǫ in Σ q X .Note that we can assume that ∠ pqr < π because (3.1) is obviously true if ∠ pqr = π . Then for s ∈ [ qr ]sufficiently close to q , there is ¯ u ∈ [¯ s ¯ p ] such that ∠ ¯ p ¯ q ¯ u = | ↑ pq ξ | (and thus ∠ ¯ s ¯ q ¯ u = | ↑ sq ξ | = | ↑ rq ξ | ),and thus u can be chosen to also satisfy | qu | = | ¯ q ¯ u | . Consequently, we can draw two triangles △ ˆ p ˆ q ˆ u and △ ˆ s ˆ q ˆ u in S k with | ˆ p ˆ q | = | pq | , | ˆ u ˆ q | = | uq | , | ˆ s ˆ q | = | sq | , ∠ ˆ p ˆ q ˆ u = | ↑ pq ↑ uq | , and ∠ ˆ s ˆ q ˆ u = | ↑ sq ↑ uq | . Notethat | ∠ ˆ p ˆ q ˆ u − ∠ ¯ p ¯ q ¯ u | ǫ and | ∠ ˆ s ˆ q ˆ u − ∠ ¯ s ¯ q ¯ u | ǫ . Then via the Law of Cosine on S k , there is a positiveconstant C such that lim sup s → q (cid:12)(cid:12)(cid:12)(cid:12) | ˆ p ˆ u | + | ˆ u ˆ s | − ( | ¯ p ¯ u | + | ¯ u ¯ s | ) | qs | (cid:12)(cid:12)(cid:12)(cid:12) Cǫ, and thus by letting ǫ → | ˆ p ˆ u | + | ˆ u ˆ s | = | ¯ p ¯ u | + | ¯ u ¯ s | + o ( | qs | ) = | ¯ p ¯ s | + o ( | qs | ) . Moreover, since both X and ¯ X are of curvature > k , by (3-2) in Subsection 1.3 we have that | pu | | ˆ p ˆ u | and | us | | ˆ u ˆ s | (3 . p, q, u ∈ ¯ X and s, q, u ∈ X ). Then by the definition of D ( X ), | ps | | pu | + | us | | ˆ p ˆ u | + | ˆ u ˆ s | = | ¯ p ¯ s | + o ( | qs | ) . In the rest of the proof, we just need to verify Claim 2. Similarly, we only consider the case where p ∈ ¯ X ◦ and q ∈ X ◦ (for other cases, (3.2) is obviously true by (3-2) on X or ¯ X ), and we can assumethat 0 < ∠ pqr < π . Then [ pq ] ∩ ∂X is a single point v (by Fact 3.2), and for the triangle △ ˜ q ˜ v ˜ s ⊂ S k with | ˜ q ˜ v | = | qv | , | ˜ q ˜ s | = | qs | and | ˜ v ˜ s | = | vs | , we have that0 < ∠ ˜ q ˜ v ˜ s | ↑ qv ↑ sv | and 0 < ∠ ˜ v ˜ q ˜ s | ↑ vq ↑ sq | (= ∠ pqr < π ) (3 . X (note that [ vq ] , [ qs ] and any [ vs ] belong to X by Fact 3.2, and there is a unique minimalgeodesic between q and v by the third possible case before Claim 1). We prolong [˜ q ˜ v ] to ˜ p with ∠ ˜ s ˜ v ˜ p = π − ∠ ˜ q ˜ v ˜ s and | ˜ v ˜ p | = | vp | . It is clear that | ˜ p ˜ s | | ¯ p ¯ s | by the second part of (3.4). Then, for s ∈ [ qr ] sufficiently close to q , it suffices to show that | ps | | ˜ p ˜ s | + o ( | qs | ) . (3 . | ↑ qv ↑ sv | + | ↑ sv ↑ pv | = π (due to | ↑ qv ↑ pv | = π in D (Σ v X )),and δ , | ↑ qv ↑ sv | → s → q (because there is a unique minimal geodesic between v and q ).And note that ↑ sv ∈ (Σ v X ) ◦ (for s close to q ) and ↑ pv ∈ (Σ v ¯ X ) ◦ , so there is ξ ∈ ∂ Σ v X such that | ↑ sv ξ | + | ξ ↑ pv | = | ↑ sv ↑ pv | = π − δ . Again by Fact 3.3, as s → q (or δ → ǫ ( δ ) → ε s → | qs | ≪ ε s such that for any positive t ε s there is u ∈ ∂X with | vu | = t such that9 ↑ uv ξ | < ǫ ( δ ). Here, a key observation is: since D (Σ v X ) is an Alexandrov space with curvature > | ↑ sv ξ | + | ξ ↑ pv | = π − δ ’ and ‘ | ↑ uv ξ | < ǫ ( δ )’ we can derive that | ↑ sv ↑ uv | + | ↑ uv ↑ pv | < π − δ + o ( δ )(note that if δ = 0, i.e. | ↑ sv ↑ pv | = π , then | ↑ sv η | + | η ↑ pv | = π for all η ∈ D (Σ v X )). Consequently, wehave a further estimate as follows: | ↑ sv ↑ uv | + | ↑ uv ↑ pv | < π − ∠ ˜ q ˜ v ˜ s + o ( | qs | ) = ∠ ˜ s ˜ v ˜ p + o ( | qs | ) . (3 . ∠ ˜ q ˜ v ˜ s | ↑ qv ↑ sv | = δ (see the first part of (3.4)) with δ → | qs | →
0, and it is easy tosee that ∠ ˜ q ˜ v ˜ s = O ( | qs | ), a positive infinitesimal of the same order as | qs | . This enables us to haveanother key observation that ∠ ˜ q ˜ v ˜ s − δ + o ( δ ) o ( | qs | ), which implies (3.6).We next let ˜ u be the point in [˜ p ˜ s ] such that ∠ ˜ p ˜ v ˜ u = | ↑ pv ξ | . Then for s ∈ [ qr ] sufficiently close to q , it also holds that | ˜ v ˜ u | = O ( | qs | ), and thus u ∈ ∂X can be chosen to also satisfy | vu | = | ˜ v ˜ u | . Weconsider △ ˜ p ˜ v ˆ u and △ ˜ s ˜ v ˆ u in S k with | ˆ u ˜ v | = | uv | , ∠ ˜ p ˜ v ˆ u = | ↑ pv ↑ uv | , and ∠ ˜ s ˜ v ˆ u = ∠ ˜ s ˜ v ˜ p − ∠ ˜ p ˜ v ˆ u whichimplies | ↑ sv ↑ uv | ˜ s ˜ v ˜ p − ∠ ˜ p ˜ v ˆ u + o ( | qs | ) = ∠ ˜ s ˜ v ˆ u + o ( | qs | ) (by (3.6)). Similar to (3.3), we have that | pu | | ˜ p ˆ u | and | us | | ˆ u ˜ s | + o ( | qs | ) . Moreover, note that ∠ ˜ u ˜ v ˆ u = | ∠ ˜ p ˜ v ˜ u − ∠ ˜ p ˜ v ˆ u | = || ↑ pv ξ | − | ↑ pv ↑ uv || | ↑ uv ξ | < ǫ ( δ ), and | ˜ u ˆ u | as aninfinitesimal of | qs | has the same order as | ˜ v ˜ u | · sin ∠ ˜ u ˜ v ˆ u (note that | ˜ v ˜ u | = | ˜ v ˆ u | = O ( | qs | ), and ǫ ( δ ) → | qs | → | ˜ u ˆ u | = o ( | qs | ) . This enables us to see that | ˜ p ˆ u | + | ˆ u ˜ s | = | ˜ p ˜ s | + o ( | qs | ) in △ ˜ p ˜ s ˆ u (a very narrow triangle). Then, bythe definition of D ( X ), | ps | | pu | + | us | | ˜ p ˆ u | + | ˆ u ˜ s | + o ( | qs | ) = | ˜ p ˜ s | + o ( | qs | ) . I.e., (3.5) is verified, and thus the proof is completed.
Appendix On f ′′ ( t ) + kf ( t ) ( > ) 0 Definition A.1 (cf. [PP]). Let f : ( a, b ) → R be a continuous function. At t ∈ ( a, b ), we say that f ′′ ( t ) ( > ) B in the support sense if there is an A ∈ R such that f ( t + τ ) ( > ) f ( t ) + Aτ + 12 Bτ + o (cid:0) τ (cid:1) . It is obvious that f ′′ ( t ) > f ( t ) achieves a local maximum orminimum at t respectively. And it might be possible that ‘ f ′′ ( t ) B ’ and ‘ f ′′ ( t ) > B ’ in thesupport sense occur simultaneously (which implies that f has the first derivative at t ), but f doesnot have the second derivative at t at all.We would like to point out that one can define f ′′ ( t ) ( > ) B in the support sense by a bit weakerconditions (refer to § § . Anyway, the following proposition always holdsin each version of the definition. In [Na] and [Pet], the definition is given respectively by: the lim sup τ → Å lim inf τ → ã of f ( t + τ )+ f ( t − τ ) − f ( t ) τ is ( > ) B ; for any ǫ >
0, there is a twice differentiable function f ǫ defined on ( t − δ, t + δ ) with some δ > f ( t ) = f ǫ ( t ) , f ( t + τ ) ( > ) f ǫ ( t + τ ) for all τ ∈ ( − δ, δ ), and f ′′ ǫ ( t ) B + ǫ ( > B − ǫ ). roposition A.2. Let f : [0 , l ] → R be a continuous function with f (0) = f ( l ) = 0 , and let k be aconstant real number such that l < π √ k if k > . If f ′′ ( t ) + kf ( t ) ( > ) 0 in the support sense for all t ∈ (0 , l ) , (A . then f ( t ) > ( ) 0 for all t ∈ [0 , l ] . Remark A.3.
Note that Proposition A.2 for k = 0 is equivalent to the following well known result: A continuous function f ( t ) | [ a,b ] is concave or convex if f ′′ ( t ) or > in the support sense for all t ∈ ( a, b ) respectively ([Na]).It turns out that the proof of Proposition A.2 for cases where k = 0 can be reduced to the casewhere k = 0. So, for cases where k = 0, Proposition A.2 should also be known to experts. Anyway,we would like to present a proof for each case. Proof of Proposition
A.2 . Note that it suffices to show that f ( t ) > ” holds in (A.1).Case 1: k = 0. In this case, f ′′ ( t ) t ∈ (0 , l ) in the support sense. Consider g ǫ ( t ) , f ( t ) − ǫ t ( t − l )where ǫ > g ǫ ( t ) is also continuous on [0 , l ] with g ǫ (0) = g ǫ ( l ) = 0, and g ′′ ǫ ( t ) − ǫ for all t ∈ (0 , l ) in the support sense. Observe that g ǫ ( t ) > t ∈ [0 , l ]; otherwise, g ǫ ( t ) will achieve its (negative) minimum at some t ∈ (0 , l ), and thus g ′′ ǫ ( t ) > g ′′ ǫ ( t ) − ǫ ’. It then follows that f ( t ) > t ∈ [0 , l ] when letting ǫ → k <
0. If f ( t ) < t ∈ (0 , l ), by the continuity there is [ a, b ] ⊆ [0 , l ] such that f ( a ) = f ( b ) = 0 and f ( t ) t ∈ [ a, b ], and thus f ′′ ( t ) − kf ( t ) t ∈ [ a, b ] (by (A.1)).This is impossible by Case 1, i.e., it has to hold that f ( t ) > t ∈ [0 , l ].Case 3: k >
0. Due to the similarity, we just give a proof for k = 1. We argue by contradiction.Suppose that f achieves its negative minimum at t ∈ (0 , l ). By the continuity of f , without loss ofgenerality, we can assume that f ( t ) t ∈ [0 , l ]. Note that, for a positive number λ , λf ( t )still satisfies (A.1); so we can assume that f ( t ) = − l √
2. Note that we can assumethat l − t t (or vice versa), so l − t √
2. For t ∈ [ t , l ], we set h ( t ) , l − t ) ( t − t ) − g ( t ) , f ( t ) − h ( t ) . Note that g ( t ) = g ( l ) = 0. And since h ′′ ( t ) = l − t ) >
1, we have that, in the support sense, g ′′ ( t ) = f ′′ ( t ) − h ′′ ( t ) − f ( t ) − − f ( t ) − . By Remark A.3, g ( t ) is a concave function on [ t , l ], so its right derivative g ′ + ( t ) exists. Moreover,note that ‘ f ( t ) = min { f ( t ) | t ∈ [0 , l ] } ’ and ‘ f ′′ ( t ) − f ( t ) = 1’ together imply that f ( t + τ ) f ( t ) + τ + o (cid:0) τ (cid:1) . It therefore follows that g ′ + ( t )
0, which together with g ( t ) = g ( l ) = 0 andthe concavity of g ( t ) implies that g ( t ) ≡ t , l ]. This is impossible because g ′′ ( t ) − f ( t ) − < t close to l .We now can assume that 2 √ < l < π , and still assume that l − t t (so l − t < π ). Set h ( t ) , − sin( t − t + π g ( t ) , f ( t ) − h ( t ) , t ∈ [ t , l ] . When k > l = π √ k , one can consider − sin( √ kt ) or sin( √ kt ) as counterexamples respectively. g ( t ) = 0 and g ( l ) >
0, and g ( t ) also satisfies (A.1) (i.e. g ′′ ( t ) + g ( t )
0) on [ t , l ]. Viathe special case above (note that l − t < π < √ g ( t ) > f ( t ) > h ( t ))on [ t , l ], so g ′′ ( t ) = f ′′ ( t ) − h ′′ ( t ) − f ( t ) + h ( t )
0. Then g ( t ) is a concave function on [ t , l ].Moreover, we similarly have that g ′ + ( t )
0. Together with g ( t ) = 0, these imply g ( t ) t , l ],which contradicts g ( l ) > Remark A.4.
Proposition A.2 still holds if f ( t ) just satisfies a piecewise version of (A.1); namely, f ′′ ( t ) + kf ( t ) ( > ) 0 in the support sense for all but at most a finite number of t ∈ (0 , l ), and f + , max ( t ) f − , min ( t ) ( f + , min ( t ) > f − , max ( t )) at each exceptional t . This is obvious in the case where k = 0 because, due to Remark A.3, f ( t ) is also concave (convex) in the situation here. Then similarto Proposition A.2, all other cases can be reduced to the case where k = 0. Remark A.5.
If the function f ( t ) in Proposition A.2 has the second derivative indeed, we would liketo present an easier proof (cf. [Pet]). Consider g ( t ) , f ( t )sn k t , t ∈ (0 , l ] (note that l < π √ k if k > , and g ′ ( t ) = f ′ ( t )sn k t − f ( t )sn ′ k t sn k t , z ( t )sn k t . Note that lim t → + z ( t ) = 0 and z ′ ( t ) = ( f ′′ ( t ) + kf ( t ))sn k t z ( t ) g ′ ( t ) , l ]. And thus, g ( t ) > g ( l ) = 0, so does f ( t ). References [AKP] S. Alexander, V. Kapovitch and A. Petrunin,
Alexandrov Geometry , 2010.[BGP] Yu. Burago, M. Gromov and G. Perel’man,
A.D. Alexandrov spaces with curvature boundedbelow , Uspeckhi Mat. Nank, 1992, 47(2): 3-51.[MV] A.D. Miller and R. Vyborny,
Some remarks on functions with one-sided derivatives , TheAmerican Mathematical Monthly, 1986, 93(6): 471-475.[Na] I.P. Natanson,
Theory of funcions of a real variable , Teoria functsiy veshchestvennoy pere-mennoy. Frederick Ungar Pub. Co, 1960.[Pe] G. Perel’man,
A.D. Alexandrov spaces with curvature bounded below II , Preprint.[PP] G. Perel’man and A. Petrunin,
Quasigeodesics and Gradient Curves in Alexandrov spaces