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arxiv: 2407.19908 · v2 · submitted 2024-07-29 · 🧮 math.SG

Symplectic structures on the space of space curves

Martin Bauer , Sadashige Ishida , Peter W. Michor This is my paper

Pith reviewed 2026-05-23 23:04 UTC · model grok-4.3

classification 🧮 math.SG
keywords symplectic geometryshape spacespace curvesMarsden-Weinstein structureRiemannian metricsHamiltonian vector fieldsunparameterized curvesLiouville form
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The pith

Symplectic structures on the shape space of unparameterized space curves generalize the classical Marsden-Weinstein structure through integration with Riemannian metrics.

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

The paper establishes new symplectic structures on the shape space of unparameterized space curves. It does so by integrating the Liouville 1-form from the Marsden-Weinstein structure with Riemannian structures from shape analysis. This generalization allows the derivation of Hamiltonian vector fields for classical Hamiltonian functions. Readers might care because it bridges symplectic geometry and shape analysis, potentially aiding in the study of curve dynamics in three-dimensional space.

Core claim

The central claim is that symplectic structures on the shape space of unparameterized space curves can be presented by integrating the Liouville 1-form of the Marsden-Weinstein structure with Riemannian structures introduced in mathematical shape analysis, and that Hamiltonian vector fields for several classical Hamiltonian functions can be derived with respect to these new symplectic structures.

What carries the argument

The integration of the Liouville 1-form of the Marsden-Weinstein structure with Riemannian structures from shape analysis, which constructs the generalized symplectic forms on the shape space.

If this is right

  • Derivation of Hamiltonian vector fields becomes possible for classical functions under these structures.
  • The structures apply specifically to unparameterized space curves.
  • They extend the classical Marsden-Weinstein structure to incorporate shape analysis metrics.
  • New symplectic forms are well-defined on the shape space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These forms could facilitate modeling of conserved quantities in physical systems involving curves.
  • Extensions might apply to parameterized curves or other manifolds by similar integration methods.
  • Applications in computational shape analysis could benefit from the symplectic perspective for optimization or dynamics.

Load-bearing premise

The integration of the Liouville 1-form with the Riemannian structures from shape analysis produces well-defined symplectic forms on the shape space.

What would settle it

A direct computation on a specific curve, such as a circle, showing that the resulting two-form fails to be closed or non-degenerate would falsify the existence of these symplectic structures.

Figures

Figures reproduced from arXiv: 2407.19908 by Martin Bauer, Peter W. Michor, Sadashige Ishida.

Figure 1
Figure 1. Figure 1: Hamiltonian flow of H−2, the flux of a rotational vector field from Example 4.5 using Φ(ℓc) = 10ℓ −2 c (top), and the flow only with its binormal component (bottom). The red, green, and blue axes are the x, y, z axes re￾spectively. in both of our examples. Example 6.1 (Flux of a vector field). We first simulate the Hamiltonian flow for the Hamil￾tonian that is defined as the flux of a vector field through … view at source ↗
Figure 2
Figure 2. Figure 2: Hamiltonian flow of hgradΩΦ(ℓ) E with different choices of Φ(ℓ). In each row the initial curve, which is not shown, corresponds to the trefoil (11). The right-most images are the front-view of the last configurations of curves showing high symmetry for the 120-degree rotation around the z-axis. points tend to get stuck once they come closer to the origin as both the term −Dsc × c and the term 1 2 |c| 2Dsc … view at source ↗
read the original abstract

We present symplectic structures on the shape space of unparameterized space curves that generalize the classical Marsden-Weinstein structure. Our method integrates the Liouville 1-form of the Marsden-Weinstein structure with Riemannian structures that have been introduced in mathematical shape analysis. We also derive Hamiltonian vector fields for several classical Hamiltonian functions with respect to these new symplectic structures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 0 minor

Summary. The manuscript claims to construct symplectic structures on the shape space of unparameterized space curves by integrating the Liouville 1-form of the classical Marsden-Weinstein structure with Riemannian metrics from shape analysis; the resulting forms are asserted to generalize the Marsden-Weinstein structure, and the paper derives the associated Hamiltonian vector fields for several classical Hamiltonians on this space.

Significance. If the constructions are shown to produce closed, reparameterization-invariant, and non-degenerate 2-forms on the quotient, the work would usefully connect symplectic geometry with shape analysis and supply explicit Hamiltonian dynamics on curve spaces. The explicit derivation of Hamiltonian vector fields is a concrete strength that would facilitate further study.

major comments (3)
  1. [main construction section] The central construction (integrating the Marsden-Weinstein Liouville 1-form with a Riemannian metric) must be shown to yield a closed 2-form that is invariant under reparameterizations and therefore descends to the quotient; without an explicit verification of dω = 0 after descent, the claim that the result is symplectic on the shape space remains unestablished.
  2. [non-degeneracy argument] Non-degeneracy on the tangent spaces to the quotient by reparameterizations is load-bearing for the symplectic claim yet is not addressed in the abstract or indicated in the provided text; a concrete argument (e.g., via an explicit pairing or kernel computation) is required.
  3. [Hamiltonian vector fields section] The Hamiltonian vector fields are derived with respect to the new structures, but their well-definedness on the quotient depends on the same invariance and non-degeneracy properties; the derivations therefore inherit the same verification gap.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments, which highlight important gaps in the explicit verification of the symplectic properties on the quotient. We agree that these verifications are necessary to fully establish the claims and will incorporate them in a revised version.

read point-by-point responses

  1. Referee: [main construction section] The central construction (integrating the Marsden-Weinstein Liouville 1-form with a Riemannian metric) must be shown to yield a closed 2-form that is invariant under reparameterizations and therefore descends to the quotient; without an explicit verification of dω = 0 after descent, the claim that the result is symplectic on the shape space remains unestablished.

    Authors: We agree that an explicit verification is required. In the revised manuscript we will add a new subsection after the central construction that computes the exterior derivative on the quotient, verifies dω = 0, and confirms reparameterization invariance so that the form descends to the shape space. revision: yes

  2. Referee: [non-degeneracy argument] Non-degeneracy on the tangent spaces to the quotient by reparameterizations is load-bearing for the symplectic claim yet is not addressed in the abstract or indicated in the provided text; a concrete argument (e.g., via an explicit pairing or kernel computation) is required.

    Authors: We acknowledge the omission. The revision will include a dedicated paragraph providing an explicit kernel computation (or pairing argument) demonstrating that the 2-form is non-degenerate on the tangent spaces to the quotient by reparameterizations. revision: yes

  3. Referee: [Hamiltonian vector fields section] The Hamiltonian vector fields are derived with respect to the new structures, but their well-definedness on the quotient depends on the same invariance and non-degeneracy properties; the derivations therefore inherit the same verification gap.

    Authors: We agree that the well-definedness of the Hamiltonian vector fields on the quotient relies on the same properties. In the revision we will add cross-references to the new invariance and non-degeneracy arguments and explicitly state that the vector fields descend once those properties are established. revision: yes

Circularity Check

0 steps flagged

No circularity: new symplectic forms constructed from independent classical and shape-analysis inputs

full rationale

The paper constructs symplectic structures on unparameterized curve shape space by integrating the Marsden-Weinstein Liouville 1-form with Riemannian metrics from prior shape analysis. This is an explicit combination of two externally cited lines of work; the resulting 2-form is defined directly rather than fitted or renamed from the target result. No self-definitional equations, fitted-input predictions, or load-bearing self-citation chains appear in the derivation. Any author-overlap citations supply independent prior structures and do not reduce the central claim to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, background axioms, or newly postulated entities.

pith-pipeline@v0.9.0 · 5573 in / 1054 out tokens · 18469 ms · 2026-05-23T23:04:42.014483+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Implicit representations of codimension-2 submanifolds and their prequantum structure

    math.SG 2025-07 unverdicted novelty 7.0

    Complex-valued implicit representations of codimension-2 submanifolds carry a prequantum bundle whose curvature recovers the Marsden-Weinstein symplectic form on the space of submanifolds.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages · cited by 1 Pith paper

  1. [1]

    V. I. Arnold and B. A. Khesin. Topological methods in hydrodynamics, volume 125 of Applied Mathe- matical Sciences. Springer, Cham, second edition, [2021] ©2021

    work page 2021

  2. [2]

    Bauer, M

    M. Bauer, M. Bruveris, P. Harms, and P. W. Michor. Vanishing geodesic distance for the riemannian metric with geodesic equation the kdv-equation. Annals of Global Analysis and Geometry , 41:461–472, 2012

    work page 2012

  3. [3]

    Bauer, M

    M. Bauer, M. Bruveris, P. Harms, and J. Moller-Andersen. A numerical framework for sobolev metrics on the space of curves. SIAM Journal on Imaging Sciences , 10(1):47–73, 2017. 30 MARTIN BAUER, SADASHIGE ISHIDA, AND PETER W. MICHOR

    work page 2017

  4. [4]

    Bauer, M

    M. Bauer, M. Bruveris, and P. W. Michor. Overview of the geometries of shape spaces and diffeomor- phism groups. Journal of Mathematical Imaging and Vision , 50:60–97, 2014

    work page 2014

  5. [5]

    Bauer and P

    M. Bauer and P. Harms. Metrics on spaces of immersions where horizontality equals normality. Differ- ential Geometry and its Applications , 39:166–183, 2015

    work page 2015

  6. [6]

    A. I. Bobenko. Geometry II: Discrete differential geometry . 2015

    work page 2015

  7. [7]

    Cervera, F

    V. Cervera, F. Mascaro, and P. W. Michor. The action of the diffeomorphism group on the space of immersions. Differential Geometry and its Applications , 1(4):391–401, 1991

    work page 1991

  8. [8]

    Chern, F

    A. Chern, F. Kn ¨oppel, F. Pedit, and U. Pinkall. Commuting Hamiltonian Flows of Curves in Real Space Forms, volume 1 of London Mathematical Society Lecture Note Series , page 291–328. Cambridge University Press, 2020

    work page 2020

  9. [9]

    Kriegl and P

    A. Kriegl and P. Michor. The Convenient Setting of Global Analysis . Mathematical Surveys. American Mathematical Society, 1997

    work page 1997

  10. [10]

    L. Lempert. Loop spaces as complex manifolds. Journal of Differential Geometry , 38:519–543, 1993

    work page 1993

  11. [11]

    A. J. Majda and A. L. Bertozzi. Vorticity and Incompressible Flow. Cambridge Texts in Applied Math- ematics. Cambridge University Press, 2001

    work page 2001

  12. [12]

    Marsden and A

    J. Marsden and A. Weinstein. Coadjoint orbits, vortices, and clebsch variables for incompressible fluids. Physica D: Nonlinear Phenomena , 7(1):305–323, 1983

    work page 1983

  13. [13]

    P. W. Michor. Some geometric evolution equations arising as geodesic equations on groups of diffeomor- phisms including the Hamiltonian approach. In Phase space analysis of partial differential equations , volume 69 of Progr. Nonlinear Differential Equations Appl. , pages 133–215. Birkh ¨auser Boston, 2006

    work page 2006

  14. [14]

    P. W. Michor. Topics in differential geometry, volume 93 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008

    work page 2008

  15. [15]

    P. W. Michor. Manifolds of mappings for continuum mechanics. In Geometric Continuum Mechanics , volume 42 of Advances in Continuum Mechanics , pages 3–75. Birkh ¨auser Basel, 2020

    work page 2020

  16. [16]

    P. W. Michor and D. Mumford. Vanishing geodesic distance on spaces of submanifolds and diffeomor- phisms. Documenta Mathematica, 10:217–245, 2005

    work page 2005

  17. [17]

    P. W. Michor and D. Mumford. Riemannian geometries on spaces of plane curves. Journal of the European Mathematical Society, 008(1):1–48, 2006

    work page 2006

  18. [18]

    P. W. Michor and D. Mumford. An overview of the riemannian metrics on spaces of curves using the hamiltonian approach. Applied and Computational Harmonic Analysis, 23(1):74–113, 2007. Special Issue on Mathematical Imaging

    work page 2007

  19. [19]

    J. J. Millson and B. Zombro. A k ¨ahler structure on moduli space of isometric maps of a circle into euclidean space. Inventiones mathematicae, 123(1):35–60, 1996

    work page 1996

  20. [20]

    T. Needham. K ¨ahler structures on spaces of framed curves. Annals of Global Analysis and Geometry , 54(1):123–153, 2018

    work page 2018

  21. [21]

    Padilla, A

    M. Padilla, A. Chern, F. Kn ¨oppel, U. Pinkall, and P. Schr ¨oder. On bubble rings and ink chandeliers. ACM Trans. Graph., 38(4), 2019

    work page 2019

  22. [22]

    P. G. Saffman. Vortex Dynamics. Cambridge Monographs on Mechanics. Cambridge University Press, 1993

    work page 1993

  23. [23]

    J. Shah. h0-type Riemannian metrics on the space of planar curves. Quarterly of Applied Mathematics , 66(1):123–137, 2008

    work page 2008

  24. [24]

    Srivastava, E

    A. Srivastava, E. Klassen, S. H. Joshi, and I. H. Jermyn. Shape analysis of elastic curves in euclidean spaces. IEEE transactions on pattern analysis and machine intelligence , 33(7):1415–1428, 2010

    work page 2010

  25. [25]

    Srivastava and E

    A. Srivastava and E. P. Klassen. Functional and shape data analysis , volume 1. Springer, 2016

    work page 2016

  26. [26]

    Tabachnikov

    S. Tabachnikov. On the bicycle transformation and the filament equation: results and conjectures. Journal of Geometry and Physics , 115:116–123, 2017

    work page 2017

  27. [27]

    Yezzi and A

    A. Yezzi and A. Mennucci. Conformal metrics and true ”gradient flows” for curves. In Tenth IEEE International Conference on Computer Vision (ICCV’05) Volume 1 , volume 1, pages 913–919. IEEE, 2005

    work page 2005

  28. [28]

    L. Younes. Shapes and diffeomorphisms, volume 171. Springer, 2010

    work page 2010