(In)equality distance patterns and embeddability into Hilbert spaces
aa r X i v : . [ m a t h . M G ] J a n (In)equality distance patterns and embeddability into Hilbert spaces Alexandru Chirvasitu
Abstract
We show that compact Riemannian manifolds, regarded as metric spaces with their globalgeodesic distance, cannot contain a number of rigid structures such as (a) arbitrarily largeregular simplices or (b) arbitrarily long sequences of points equidistant from pairs of pointspreceding them in the sequence. All of this provides evidence that Riemannian metric spacesadmit what we term loose embeddings into finite-dimensional Euclidean spaces: continuousmaps that preserve both equality as well as inequality.We also prove a local-to-global principle for Riemannian-metric-space loose embeddability:if every finite subspace thereof is loosely embeddable into a common R N , then the metric spaceas a whole is loosely embeddable into R N in a weakened sense. Key words: Riemannian manifold; geodesic; isometry; Euclidean distance
MSC 2010: 30L05; 53B20; 53B21
Introduction
The present note is a follow-up on [3], where the following notion was introduced ([3, Definition2.2]):
Definition 0.1
Let (
X, d X ) and ( Y, d Y ) be two metric spaces. A continuous map f : X → Y is a loosely isometric (or just loose ) embedding if d X ( x, x ′ ) d Y ( f x, f x ′ )is a well-defined one-to-one map on the codomain of d X .( X, d X ) is loosely embeddable (or LE) in ( Y, d Y ) if it admits such a f : X → Y , and it is justplain LE (without specifying Y ) if it is loosely embeddable into some finite-dimensional Hilbertspace. (cid:7) In other words, f turns (un)equal distances into (un)equal distances respectively. Note inparticular that loose embeddings are automatically one-to-one, so they are embeddings. It is thecondition of being isometric that is being loosened.The concept was originally motivated by the fact that, by (a slight paraphrase of) [7, Corollary4.9], LE compact metric spaces have quantum automorphism groups , i.e. they admit universalisometric actions by compact quantum groups. Despite its origin in the non-commutative-geometryconsiderations central to [3], the notion seems to hold some independent interest of its own. Roughlyspeaking: Loose embeddability captures the combinatorial patterns of distance (in)equality achievable inHilbert spaces.
1e focus in particular on
Riemannian compact metric spaces, i.e. those obtained by equip-ping Riemannian manifolds with their global geodesic distance (Definition 1.1). According to [4,Theorem 0.2] compact connected Riemannian manifolds (with or without boundary) always havequantum isometry groups, which in fact coincide with their classical isometry groups. This meansthat in the present context we are departing from the initial motivation for considering loose em-beddability, namely the existence of quantum isometry groups. The concept nevertheless suggestssome apparently-non-trivial questions in metric and Riemannian geometry.These seem particularly well suited for loose embeddability, as attested by several results rulingout non-LE metric configurations in the Riemannian context: • Lemma 1.2 is the simple remark that 1-dimensional Riemannian manifolds are always looselyembeddable. • In Proposition 1.3 we observe that a (compact) Riemannian metric space cannot contain n -tuples of equidistant points for arbitrarily large n . • Generalizing this, Theorem 1.6 shows that compact Riemannian metric spaces do not containarbitrarily large sets of pairs { x i , y i } of points with x i and y j both equidistant from x i and y i for all j > i . This rules out (in the Riemannian case) a subtler class of counterexamplesto loose metric embeddability given by Lemma 1.5.In short, Riemannian compact metric spaces make for poor counterexamples to loose metric em-beddability.Section 2 further contains a number of questions related to loose embeddability, and answer aweakened form of Question 2.1 affirmatively in the Riemannian setting: Theorem 2.4 says, roughlyspeaking, that if the finite subspaces of a compact Riemannian metric space ( M, d ) are uniformly loosely embeddable (i.e. loosely embeddable into Hilbert spaces of uniformly bounded dimension)then (
M, d ) itself is loosely embeddable in a weak sense.
Acknowledgements
This work was partially supported by NSF grants DMS-1801011 and DMS-2001128.
It is a natural problem to determine to what extent various classes of metric spaces are loosely em-beddable in the sense of Definition 0.1. Of special interest, for instance, are Riemannian manifoldsequipped with the geodesic metric. [5, 1] are good sources for the Riemannian geometry we willperuse.
Definition 1.1 A Riemannian metric space is a connected Riemannian manifold equipped withthe global geodesic metric. (cid:7)
Unless specified otherwise, all of our metric spaces are assumed compact; we thus often dropthat adjective for brevity. Note first:
Lemma 1.2
One-dimensional compact Riemannian metric spaces are loosely embeddable.
Proof
Indeed, 1-dimensional connected compact Riemannian manifolds are isometric to the unitcircle, which can be loosely embedded into the plane via its standard origin-centered-unit-circlerealization. (cid:4)
2t is observed in [3, Example 2.3] that metric spaces containing regular n -simplices (i.e. point n -tuples with equal pairwise distances) for each n cannot be loosely embeddable. The followingobservation rules this out for Riemannian metric spaces. Proposition 1.3
Let ( M, d ) be a compact Riemannian manifold with its geodesic metric. Thereis an upper bound on the number of vertices of a regular simplex in M . Proof
Now let v and v be two other vertices of ∆, chosen so that the angle ε = ∡ v v v issufficiently small (possible for large n ). Since M is compact there is a global lower bound K for itssectional curvature. By the Toponogov comparison theorem ([1, § v v is bounded above by the length of the third edge in an isosceles triangle with angle ε subtending the two edges of equal length ℓ = v v = v v in the space form [1, § K . This length goes to 0 as ε does, contradicting v v = ℓ . (cid:4) Large regular simplices are not the only obstruction to loose embeddability. The somewhat moresophisticated configurations that pose problems involve, roughly speaking, large sets of points eachequidistant to large sets of pairs of points. To make sense of this we need some terminology.
Definition 1.4
Let n be a positive integer. An n -flag of median hyperplanes is a collection ofpoints { p i , q i , ≤ i ≤ n − } (1-1)such that d ( z, p s ) = d ( z, q s )for all z = p i or q i with i > s . (cid:7) The term ‘median hyperplane’ is meant to invoke the locus of points in a Euclidean space thatare equidistant from two given points, while ‘flag’ means chain ordered by inclusion, as in { p i , q i } i ≥ ⊃ { p i , q i } i ≥ ⊃ · · · . (1-2)The relevance of the concept stems from the following simple remark. Lemma 1.5
A compact metric space containing n -flags of median hyperplanes is not LE. Proof
If such a space (
X, d ) were loosely embeddable in R d say, then each of the sets (1-2) wouldbe contained in a hyperplane of R d , namely the median hyperplane of (the images in R d of) p i and q i . These hyperplanes would be orthogonal in the sense that their range projections commute, soany > d of them would intersect trivially. (cid:4) On the other hand, Riemannian manifolds can still not be discounted as LE on the basis ofLemma 1.5.
Theorem 1.6
A compact Riemannian manifold ( X, d ) equipped with the geodesic metric cannotcontain n -flags of median hyperplanes for arbitrarily large n . This will require some amount of preparation.First, we will have some make some size estimates (for angles, distances, etc.). This raises theusual issue of starting with quantities that are within ε > ε such as, say Cε for some constant C . In order to avoid such irrelevancieswe make the following 3 onvention 1.7 ε will typically denote a small positive real, and whenever a new small quantitydepending on ε is introduced, we denote it by decorating ε with the usual symbols used to indicatedifferentiation. So for instance ε ′ , ε ′′ , ε (5) , etc. all denote small positive reals depending on ε insome unspecified fashion.The same notational convention applies to other symbols meant to denote small positive reals. (cid:7) In the discussion below we will modify the Riemannian tensor g on a geodesic ball of a Rieman-nian manifold ( M, g ) so as to “flatten” said ball. The relevant concept is
Definition 1.8
Let B ⊂ M be a geodesic ball in a Riemannian manifold M with tensor g , andsuppose we have fixed a coordinate system for B . We say that g is ε -Euclidean to order k along B if the derivatives of orders ≤ k of g within ε of their usual Euclidean counterparts, uniformly on B , in the respective coordinate system.We typically omit k from the discussion, simply assuming it is large enough ( k ≥ ε -Euclidean .The specific ε > ε sufficiently small, and the various coordinate choices willnot affect this. (cid:7) As a consequence of the smooth dependence of ODE solutions on the initial data (e.g. [6,Theorem B.3]), “sufficiently Euclidean” Riemannian metrics in the sense of Definition 1.8 have“sufficiently straight” geodesics. More formally (keeping in mind Convention 1.7):
Proposition 1.9
Let ( M, g ) be a Riemannian manifold, ε -Euclidean with respect to some coordi-nate system. Then, for every geodesic γ in M , parallel transport of vectors along γ does not alterangles by more than ε ′ (cid:4) Notation 1.10
Let M be a Riemannian manifold with metric tensor g and geodesic distance d .We write inj( M ) := injectivity radius of M ([5, p.271] or [1, p.142, Definition 23]): the largest number such that all pairs of points less thaninj( M ) apart are joined by a unique geodesic segment.For points p, q in a Riemannian manifold M with ℓ := d ( p, q ) < inj( M )we write γ qp : [0 , ℓ ] → M for the geodesic arc from p to q , parametrized by arclength. We will also abuse notation and denotethe image of γ qp by the same symbol. (cid:7) Definition 1.11
Let p, q be points in a Riemannian manifold M , less than inj( M ) apart. The angle ∠ ( v p , v q )between two tangent vectors v p ∈ T p M and v q ∈ T q M is defined by • parallel-transporting ([5, Chapter 2, Proposition 2.6 and Definition 2.5] or [1, p.264, Propo-sition 61]) the unit velocity vector v q to a vector v ∈ T p M along γ qp ;4 set ∠ ( v p , v q ) := angle between v p and v, computed in T p ( M ) as usual, via the Riemannian tensor.For points p , q , p ′ , q ′ in M , each two less than inj( M ) apart, the angle ∠ ( γ qp , γ q ′ p ′ ) is the angle(defined as above) between the unit velocity vectors ( γ qp ) ′ (0) and ( γ q ′ p ′ ) ′ (0). (cid:7) Remark 1.12
Although Definition 1.11 appears to bias one of the pairs p, q and p ′ , q ′ over theother, the notion is in fact symmetric: because parallel transport is an isometry between tangentspaces, whether we parallel-transport ( γ q ′ p ′ ) ′ (0) to T p M or ( γ qp ) ′ (0) to T p ′ M does not affect the value of the angle. (cid:7) For an n -dimensional Riemannian metric space ( M, d = d M ) with a basepoint z ∈ M we willconsider small geodesic balls B r = B r ( z ) := { q ∈ M | d ( z, q ) ≤ r } centered at z , parametrized with normal coordinates [1, § x i , 1 ≤ i ≤ n (so z is identified withthe origin (0 , · · · , z are identified withstraight line segments.Having fixed such a coordinate system, we can speak about segments in B , angles betweenthose segments, etc.; it will be clear from context when these are actual segments in the ambient R n housing B rather than, say, geodesic segments in M .Typically, the radius r decorating B r will be small. We will occasionally have to normalize theRiemannian metric in B r , scaling distances from the origin z = 0 ∈ B by r so that the new ball n B r (‘n’ for ‘normalized’) has radius 1.This normalization procedure has the effect of “flattening” the Riemannian metric, in the sensethat the Riemannian structure can be made arbitrarily ε -Euclidean (Definition 1.8) as r → M with geodesic metric d = d M , we write η ( p, q ) = η M ( p, q ) := d ( x, y ) (1-3)for the squared-distance function (the notation matches that in [10] for instance, where this functionfeatures prominently). Lemma 1.13
Let M be a Riemannian manifold and B = B r ( z ) a sufficiently small geodesic ballequipped with normal coordinates around z ∈ M . Let also p ∈ B be a point and consider thefunction ψ : x η ( x, p ) . with η as in (1-3). Denoting by v ∈ T z M the unit vector tangent to the geodesic z → p , the gradient ∇ ψ at z equals − d ( z, p ) v . roof This is immediate after choosing a normal coordinate system around p , whereupon ψ be-comes ψ : ( x , · · · , x n ) → n X i =1 ( x i ) . (cid:4) Proof of Theorem 1.6
Suppose we do have arbitrarily large flags of median hyperplanes in ourcompact Riemannian space ( M, d ). Since M is compact, we can assume that some large flag (1-1)is contained entirely within some small geodesic ball B r centered at a point z := p n constitutingthe flag.We can assume r is small enough that the normalized ball n B r is ε -Euclidean in the sense ofDefinition 1.8. Furthermore, because the size of the flag can also be chosen arbitrarily large, wecan also assume that ∠ (cid:0) γ q i p i , γ q j p j (cid:1) < ε ′ , ∀ ≤ i = j < n Henceforth, it will be enough to work with p i and q i for i = 0 ,
1. By the flag condition, both p and q are equidistant from p and q . Additionally, we have ∠ (cid:0) γ q p , γ q p (cid:1) < ε ′ (1-4)Furthermore, because the metric is ε -Euclidean, the unit-length velocity vectors v x along γ := γ q p stay within an angle of ε ′′ of the initial velocity vector ( γ q p ) ′ (0) (by Proposition 1.9), so (1-4) impliesthat ∠ (cid:0) v x , ( γ q p ) ′ (0) (cid:1) < ε (3) , ∀ x ∈ γ q p . (1-5)For each x ∈ γ , we saw in Lemma 1.13 that the gradient of the function ψ : x d ( x, p ) − d ( x, q ) (1-6)is 2 d ( x, q )( γ qx ) ′ (0) − d ( x, p )( γ px ) ′ (0) . (1-7)This is the parallel transport of 2 −−→ p q to x in the usual, Euclidean metric, so by our assumptionthat the original metric is ε -Euclidean the angle between (1-7) and ( γ q p ) ′ (0) is < ε (4) . To summarize,we have • a small angle between each gradient ∇ x ψ of ψ along γ , given by (1-7), and ( γ q p ) ′ (0); • a small angle between the latter and the unit tangent vectors v x at x ∈ γ , by (1-5).In particular, at each x along γ the gradient ∇ x ψ and the velocity along γ have positive inner prod-uct. This means that the function ψ in (1-6) increases strictly along the geodesic γ , contradictingthe fact that it must take the value 0 at both endpoints p and q . (cid:4) Questions
Proposition 1.3 and Theorem 1.6 seem to suggest that compact Riemannian metric spaces areparticularly amenable to loose metric embeddability. I do not know whether they are always LE,but that problem decomposes naturally: first,
Question 2.1
Let ( X, d ) be a compact metric space and N ∈ Z > a positive integer such that everyfinite subspace of ( X, d ) is loosely embeddable into R N . Does it follow that ( X, d ) itself is LE? In other words, does uniform loose embeddability for the finite subspaces of (
X, d ) entail theLE property for X as a whole?Secondly, to circle back to the Riemannian context: Question 2.2
Do compact Riemannian metric spaces satisfy the hypothesis of Question 2.1?
We conclude with a partial answer to Question 2.1. First, we need
Definition 2.3
A metric space (
X, d X ) is weakly loosely embeddable (or weakly LE ) in the metricspace ( Y, d Y ) if there is an injective map f : X → Y satisfying only the backwards implication inthe biconditional implicit in Definition 0.1: d Y ( f x, f x ′ ) = d Y ( f z, f z ′ ) ⇐ d X ( x, x ′ ) = d X ( z, z ′ ) . (2-1) (cid:7) Theorem 2.4
Under the hypotheses of Question 2.1, a compact Riemannian metric space is weaklyLE in R N . Proof
Let (
M, d ) be a compact Riemannian manifold with its geodesic metric and denote by ( F , ⊆ )the poset of finite subsets F ⊂ M (ordered by inclusion). For each F ∈ F we fix a map ψ F : F → B := origin-centered unit ball in R N such that • ψ F is a loose embedding of ( F, d ), rescaled if needed so as to ensure it lands in the ball B ; • the diameter of ψ F ( F ) is precisely 1, with ψ F p = 0 and ψ F q on the unit sphere ∂B for some p, q ∈ F .This gives us an F -indexed net [9, Chapter 3, p.187] ψ F of maps F → B , and since • B is compact; • every element p ∈ M belongs to sufficiently large F ∈ F , i.e. to the upward-directed set { F ∈ F | p ∈ F } , we can take the pointwise limit ψ ( p ) := lim F ψ F ( p ) ∈ B to obtain a map ψ : M → B . it remains to prove that ψ (a) satisfies the weak LE condition (2-1); 7b) is continuous;(c) is one-to-one. (a): condition (2-1). We want to prove that | ψx − ψx ′ | = | ψz − ψz ′ | ⇐ d M ( x, x ′ ) = d M ( z, z ′ ) (2-2)holds; this follows by passing to the limit over F ∈ F in the analogous implication for the partially-defined maps ψ F : F → B .We can now define a map ϕ : (set of distances d M ( p, q )) → R ≥ (2-3)by ϕ ( d M ( p, q )) = | ψp − ψq | . (2-4)We define the maps ϕ F , F ∈ F similarly, substituting ψ F for ψ in (2-4). (b): ψ is continuous. We have to argue thatlim d → ϕ ( d ) = 0 . If not, we can find a subnet ( F α ) α of F and points p α , q α ∈ F α such that d M ( p α , q α ) → ε := inf α | ψ F α p α − ψ F α q α | >
0; (2-5)we abbreviate ψ α := ψ F α , and similarly for ϕ .If ℓ > M , for instance [2, Definitionfollowing Theorem III.2.3]), then ( M, d M ) contains geodesic triangles with edges ℓ, ℓ, t for every 2 ℓ > t >
0. This can easily be seen, for instance, by continuously decreasing the anglebetween two length- ℓ geodesic rays based at a point from π to 0; the distance between the extremitiesof those geodesic rays will then decrease continuously from 2 ℓ to 0.Now fix some ℓ >
0, sufficiently small. We will have d M ( p α , q α ) < ℓ for sufficiently large α , andhence, by the preceding remark, we can find geodesic triangles in M with edges ℓ , ℓ and d M ( p α , q α )(assuming also that α is large enough to ensure that F α contains the tip of that isosceles geodesictriangle).Applying ψ α , we have a triangle in B with edges ϕ α ( ℓ ) , ϕ α ( ℓ ) , ϕ α ( d M ( p α , q α )) . In particular, we have ϕ α ( ℓ ) ≥ ϕ α ( d M ( p α , q α ))2 ≥ ε > ℓ > α we can find arbitrarily large finite subsets F of M , of girth ≥ ℓ (i.e. so that allpairs of points are at least ℓ apart), and hence so that (by (2-6)) | ψp − ψq | ≥ ε , ∀ p, q ∈ F. Since the cardinality of F (and hence that of ψ ( F )) can be made arbitrarily large, we are contra-dicting the compactness of B . This completes the proof of (b) above. (c): ψ is injective. Suppose not. In a sense, this means we are in precisely the oppositesituation to that encountered in the proof of part (b): there is a subnet ( F α ) α of F with points p α , q α ∈ F α such that ℓ := inf α d M ( p α , q α ) > α | ψ α p α − ψ α q α | = 0 . (2-7)By compactness, we can also assume p α and q α are convergent and hence in particular that thedistances ℓ α := d M ( p α , q α )are as well. For sufficiently small t > M with edges ℓ α , ℓ α , t for all α (consider two geodesic rays of length ℓ α with common origin, subtending small angles atsaid origin). But then an application of one of the ψ α will yield a triangle with edges ϕ α ( ℓ α ) , ϕ α ( ℓ α ) , ϕ α ( t )with ϕ and ϕ α := ϕ F α as in (2-4) and subsequent discussion, meaning that ϕ α ( t ) ≤ ϕ α ( ℓ α )2 . Since the right hand side converges to zero by (2-7), we conclude that ϕ ( t ) = 0 for all sufficientlysmall t >
0. In other words, ψ identifies any two points that are sufficiently close. (2-8)Now, for each α we also have, by assumption, points x α , y α in F α that achieve distance 1 uponapplying ψ α : | ψ α x α − ψ α y α | = 1By compactness, passage to a subnet if necessary allows us to assume that x α and y α converge to x and y in M respectively, and limiting over α produces | ψx − ψy | = 1.For some distance t > N of length- t segments x =: p → p , p → p , · · · , p N − → p N := y connecting x and y . Applying ψ we similarly obtain a broken geodesic consisting of N length- ϕ ( t )segments connecting x and y , but the latter are distance 1 apart while ϕ ( t ) = 0 by (2-8). Thisgives the contradiction we seek and finishes the proof. (cid:4) emark 2.5 As noted above, compact connected Riemannian manifolds are known to have quan-tum isometry groups and the latter are classical. We note however that the results proven above forRiemannian manifolds involve only local considerations. This means that, for instance, Proposition 1.3and theorems 1.6 and 2.4 apply to submanifolds with corners of compact Riemannian manifolds.Recall [8, Definition 2.1] that the latter are manifolds modeled as usual, via atlases, on thespaces R k ≥ × R n − k . One can obtain interesting metric spaces by • starting with a Riemannian manifold; • cutting out a domain bounded by hypersurfaces intersecting transversally ; • restricting the global geodesic metric to that domain.As soon as one goes beyond manifolds with boundary such spaces are not covered by the mainresults of [4]. (cid:7) References [1] Marcel Berger.
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Department of Mathematics, University at Buffalo, Buffalo, NY 14260-2900, USA
E-mail address : [email protected]@buffalo.edu