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Invariant distances on spaces of Legendrians

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper constructs unbounded invariant distances on universal covers of Legendrian isotopy classes using only a positive Legendrian loop, not a positive contactomorphism loop, and proves all invariant distances on Legendrian classes…

desk verdict A genuinely useful paper: it removes the positive-contact-loop assumption for unbounded invariant Legendrian distances, with the main caveat being a black-boxed finiteness result from the author's earlier work. read the letter →

arxiv 2507.18724 v1 pith:5I3PU43S submitted 2025-07-24 math.SG math.DG

Pierre-Alexandre Arlove This is my paper · ORCID
classification math.SGmath.DG MSC 53D1053D3557R17
keywords Legendrianisotopyinvariantdistanceuniversalcoverorderabilitypositiveloopcontactomorphismspectralcontactrigidity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that an unbounded invariant distance can exist on the universal cover of a Legendrian isotopy class without requiring a positive loop of contactomorphisms. It is enough to have an orderable space $\tilde L(\Lambda_*)$ and a positive loop of Legendrians that extends to a (not necessarily positive) loop of contactomorphisms. The distance is built by counting how many whole turns of this Legendrian loop separate two lifts in the order relation, and the proof shows the counting functions are finite integers. The same paper proves a complementary rigidity statement: every invariant distance on a Legendrian isotopy class is strongly discrete, so invariant distances on these spaces cannot have accumulating values. The concrete example of the unitary cotangent of the torus gives the first contact manifold whose contactomorphism group is orderable yet whose Legendrian space carries an unbounded invariant distance.

What carries the argument

The load-bearing object is the pair of counting functions $\ell_\pm$ defined through the non-negative Legendrian order $\preceq$ and the integer shifts $\tilde\varphi^N$ of the extended positive Legendrian loop. These functions measure, in whole turns of the loop, how far two lifts of Legendrian paths can be separated in the order; orderability guarantees the numbers are finite, and the extension to a contactomorphism loop makes $\tilde\varphi^N$ central, which gives the distance its invariance under the full universal cover of the contactomorphism group. The distance is then the max of $\ell_+$ and $-\ell_-$, and the unboundedness comes from the fact that $\ell_+(\tilde\Lambda^T_*,\tilde\Lambda_*) = \lceil T\rceil$ along the loop.

What would settle it

In the torus example, take the Reeb-flow loop and check directly whether $\tilde\Lambda^{1/2}_* \preceq \tilde\Lambda^1_*$ and $\tilde\Lambda^{-1}_* \preceq \tilde\Lambda^{1/2}_*$ in the non-negative Legendrian order. The proof of Proposition 4.2 forces these order relations to hold and forces $\ell_+(\tilde\Lambda^{1/2}_*,\tilde\Lambda_*)=1$; if any such order inequality fails for some real $T$, the integer-valued finiteness of $\ell_\pm$, and with it the unbounded distance of Theorem 4.1, would be false.

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Extended reading notes

Core claim

The central discovery is a construction of an unbounded invariant distance on $\tilde L(\Lambda_*)$ from order data alone. Under the assumptions that $\tilde L(\Lambda_*)$ is orderable and that a positive Legendrian loop $(\Lambda^t_*)$ can be extended to a loop of contactomorphisms $(\varphi^t)$ based at the identity, the paper defines $\ell_+(\tilde\Lambda_1,\tilde\Lambda_0) = \inf\{N\in\mathbb Z\mid \tilde\Lambda_1\preceq \tilde\varphi^N\cdot\tilde\Lambda_0\}$ and $\ell_-$ by the corresponding supremum, and proves these are integers with $\ell_-\le \ell_+$. Then $d(\tilde\Lambda_0,\tilde\Lambda_1)=\max\{\ell_+(\tilde\Lambda_1,\tilde\Lambda_0),-\ell_-(\tilde\Lambda_1,\tilde\Lambda_0)\}$ is a genuine invariant distance, compatible with the order, and unbounded because $d(\tilde\Lambda^T_*,\tilde\Lambda_*)\ge T$ for all $T>0$. The proof relies on the centrality of the integer shifts $\tilde\varphi^N$ in the universal cover of the contactomorphism group, which is exactly where the extension condition is used. As a corollary, the unitary cotangent bundle of the $n$-torus yields an orderable contactomorphism group with an unbounded invariant distance on $\tilde L$, the first such example.

Load-bearing premise

The construction rests on the assumption that the positive Legendrian loop can be extended to a loop of contactomorphisms based at the identity; without that extension the integer shifts $\tilde\varphi^N$ that the distance counts are not available, and the finiteness and unboundedness arguments collapse.

Editorial extensions

If this is right

  • Any orderable Legendrian class containing a positive Legendrian loop that extends to a contactomorphism loop admits an unbounded invariant distance given by the explicit max formula.
  • The unitary cotangent of the $n$-torus is the first example where the contactomorphism group has no positive loop but the Legendrian universal cover still carries an unbounded invariant distance.
  • Every invariant distance on a Legendrian isotopy class is strongly discrete, and on the universal cover any two lifts over different Legendrians are uniformly separated, so such distances cannot have values accumulating toward zero.
  • The constructed distance is compatible with the order relation and is quasi-isometric to the invariantized spectral pseudo-distance, so it encodes the same Reeb-dynamical information as spectral selectors up to bilipschitz equivalence.
  • The examples suggest a route to unbounded conjugation-invariant norms on the universal cover of the contactomorphism group even when that group is orderable, provided certain spectral bounds hold.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An extension beyond the paper: the extension condition might be relaxable; if integer shifts could be defined from a Legendrian loop alone, the same formula would likely produce unbounded distances in orderable classes without any contactomorphism extension.
  • An extension beyond the paper: Theorem B's strong discreteness is probably a general contact-flexibility phenomenon, so one would expect any invariant distance on homogeneous spaces built from local Weinstein neighborhoods to be strongly discrete by the same squeezing argument.
  • An extension beyond the paper: the quasi-isometry with the spectral pseudo-distance suggests that the new distance is a discretized displacement spectrum; computing it on lifts separated by Reeb flow times could yield a numerical invariant of the contact form.
  • An extension beyond the paper: the paper conjectures that smooth orbit spaces of submanifolds of high codimension admit no unbounded invariant distance; a testable route would be to construct explicit bounded bi-invariant metrics on diffeomorphism groups.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper constructs new unbounded invariant distances on the universal cover ~L(Λ*) of a Legendrian isotopy class, under the assumptions that ~L(Λ*) is orderable and that there exists a positive loop of Legendrians extending to a loop of contactomorphisms based at the identity. The key novelty is that the loop of contactomorphisms is not required to be positive. The distance is defined via integer-valued spectral selectors ℓ± that count turns of the positive Legendrian loop in the partial order. A corollary gives the first example of an orderable contact manifold for which ~L admits an unbounded invariant distance despite the absence of positive loops of contactomorphisms. The paper also proves Theorem B: every invariant distance on L(Λ*) is strongly discrete, and on ~L(Λ*) distinct underlying Legendrians are uniformly separated, via contact flexibility techniques. Additional results compare the new distance with the spectral distance and the Colin–Sandon oscillation distance, and an appendix identifies the set-theoretic universal cover with the topological one.

Significance. If the main theorem holds, it is a meaningful advance: it removes the standard assumption of a positive loop of contactomorphisms, replacing it with the weaker condition of a positive Legendrian loop admitting an arbitrary contact extension. The corollary on the unit cotangent bundle of the torus is striking, as it exhibits an orderable contact manifold with an unbounded invariant distance on ~L. Theorem B is a clean rigidity result, and its proof via contact flexibility is elegant. The paper also provides a full proof of the Weinstein neighborhood theorem (Lemma 3.5) and of the identification of the universal cover (Appendix), which are useful contributions in their own right. The writing is generally clear, and the main constructions are explicit and checkable.

major comments (2)
  1. [Section 4, Proposition 4.2] The proof of finiteness of ℓ± depends on [2, Proposition 3.1], quoted to assert that the real-valued selectors ℓ^{Λ*}_± take finite values for the family ~Λ^T_* generated by the loop (φ^t). However, the hypotheses of that proposition are not stated. It is not clear whether [2, Prop 3.1] applies to a family generated by a loop of contactomorphisms that is not positive; if it requires a positive loop of contactomorphisms, the finiteness step in Proposition 4.2 lacks support. Since finiteness of ℓ± is load-bearing for Theorem A, the authors must either state the exact hypotheses of [2, Prop 3.1] and verify them for the family ~Λ^T_*, or provide a direct proof of the finiteness.
  2. [Section 4, Propositions 4.2 and 4.3] The proof of Proposition 4.2 contains reversed inequalities: from ~φ^{-N}·~Λ0 ⪯ ~Λ1 ⪯ ~φ^N·~Λ0 the correct conclusions are ℓ+(~Λ1,~Λ0) ≤ N and ℓ-(~Λ1,~Λ0) ≥ -N, not the reverse as printed. A further argument using monotonicity of the family ~φ^N·~Λ0 and antisymmetry is needed to rule out ℓ+ = -∞ and ℓ- = +∞. The proof of Proposition 4.3 also has an incorrect first equality: it should read ℓ+(~Λ1,~Λ0) = inf{N | ~Λ1 ⪯ ~φ^N·~Λ0}, with the subsequent equivalences adjusted accordingly. These are not mere typographical slips in peripheral material; they appear in the proof of the central distance properties, and the main theorem is not fully established until they are corrected.
minor comments (4)
  1. [Proof of the Corollary, Section 1.1] The notation [t] is used ambiguously, both for the class of t in S^1 = R/Z and for the corresponding coordinate in T^n = R^n/Z^n. Please clarify the notation to avoid confusion in the definition of the loop φ^t.
  2. [Section 5.1, proof of Theorem 5.1] The inequality ~φ^{Tm}_α·~Λ* ⪯ ~φ^T·~Λ* for T ≥ 0 (and its reverse for T ≤ 0) is stated without proof or reference. Since this is used to compare the spectral distance with the new distance, a short justification or a citation to a known comparison principle would be helpful.
  3. [Throughout] There are several typographical errors: 'Riemaniann metric' in Lemma 3.5, 'concanated' in the Appendix, and the running title appears as 'INV ARIANT' in the arXiv source. These should be corrected in the final version.
  4. [Section 5.2, Proposition 5.5] In the proof of Proposition 5.5, the choice k0 := d(~Λ,~Λ') is valid because d is integer-valued, but the step k0A could be made more explicit by noting that dCS,osc(~Λ, ~φ^k·~Λ) ≤ kA for k = 2k0 via the triangle inequality; the current text is terse but correct.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the distance is explicitly constructed from the order and the given positive Legendrian loop, unboundedness is computed directly along that loop, and the main external input [2, Prop. 3.1] is a prior parameter-free theorem, not an assumption of the conclusion.

full rationale

The paper's Theorem A is a construction, not a prediction. The distance d is defined as max{ℓ+, -ℓ-} where ℓ± count integer powers of the given loop in the order relation; symmetry, triangle inequality, invariance, and finiteness are proved from the order axioms and the centrality of the integer deck transformations ~φ^N (Propositions 4.2–4.5). Unboundedness is a direct calculation ℓ+(~Λ^T_*, ~Λ_*) = ceil(T) along the very loop used in the definition, which is legitimate because the loop is assumed positive: it is not a fitted quantity masquerading as a prediction. The only externally imported finiteness result is [2, Prop. 3.1], cited verbatim; although [2] shares an author, it is a separately published, parameter-free statement whose hypotheses (orderability and a positive family) do not include the target conclusion, so it functions as genuine evidence rather than a circular self-citation. Theorem B is proved from standard flexibility facts (Weinstein neighborhoods and squeezing) and does not feed back into Theorem A. No equation is observed to reduce to its own input, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

All nontrivial inputs are listed. No free parameters are fitted to data; the construction is parameter-free. The central claim rests on the stated hypotheses of Theorem A plus standard flexibility theorems and the cited spectral-selector finiteness from [2].

assumptions (7)
  • domain assumption ~L(Λ*) is orderable, meaning the non-negative Legendrian order is antisymmetric and no positive Legendrian loop exists.
    Hypothesis of Theorem A; used through finiteness of spectral selectors and antisymmetry of the order, especially in Proposition 4.2.
  • domain assumption There is a positive Legendrian loop (Λ^t_*) based at Λ* that can be extended to a loop of contactomorphisms (φ^t) based at the identity.
    Hypothesis of Theorem A; provides the central integer shifts ~φ^N and the comparison family used to define d.
  • standard math Finiteness and order-compatibility of the real spectral selectors ℓ± stated in [2, Prop 3.1] for orderable ~L.
    External published theorem from the author's earlier joint paper; gives the bounds needed to prove that ℓ± are finite. It is cited, not reproved.
  • standard math Weinstein neighborhood theorem: a closed Legendrian has a contact neighborhood contactomorphic to its 1-jet bundle, as in Proposition 3.1.
    Used in Theorem B; the proof appeals to [24] and Lemma 3.5.
  • standard math Contact squeezing in the 1-jet bundle: a set B in a compact subset of π^{-1}(D) can be isotoped into any open neighborhood B' of an open ball D' in Λ* while fixing the zero-section, as in Proposition 3.2.
    Proved using the Liouville flow; the proof relies on Lemma 3.4, which is left to the reader.
  • standard math Elements of π1 of the identity component of the contactomorphism group are central in its universal cover, so ~φ^N commutes with other lifts.
    Standard fact about universal covers of topological groups; used repeatedly in Propositions 4.2 and 4.5.
  • standard math For the torus example, ~L is orderable by [9, Theorem 1.11] and Cont0 is orderable by [12].
    Used in the Corollary to exhibit the first example with an unbounded invariant distance but no positive contact loop.

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Pith. "Pith review of Invariant distances on spaces of Legendrians." pith.science (2026). https://pith.science/paper/5I3PU43S

@misc{pith2026250718724,
  author       = {Pith},
  title        = {Pith review of: Invariant distances on spaces of Legendrians},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5I3PU43S}},
  note         = {Machine review of arXiv:2507.18724}
}
read the original abstract

We construct new unbounded invariant distances on the universal cover of certain Legendrian isotopy classes. This is the first instance where unboundedness of an invariant distance is obtained without assuming the existence of a positive loop of contactomorphisms. We also show that invariant distances on Legendrian isotopy classes have to be discrete.

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