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Spectral resolution and rigidity of spherical space forms

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Abstract

Let n≥5n\ge5 be odd. We prove quantitative rigidity from scalar Laplace spectra for two classes of round nn-manifolds: homogeneous spherical space forms of even order and scalar lens spaces of odd order at least five. At fixed sufficiently small normalized frequency tolerance, an explicit strict bound on the lower limit of the least total unmatched multiplicity divided by R(n+1)/2R^{(n+1)/2}, along prescribed cutoffs RR, forces a closed connected Riemannian manifold to be isometric to the reference space. Nonisometric lens spaces show that this exponent is sharp in general. A Besse homotopy nn-sphere is round if it admits a constant-displacement isometry of finite odd order at least five.

Let PP be a self-adjoint classical elliptic operator of order one with positive scalar principal symbol, acting on a Hermitian bundle of positive rank over a closed connected dd-manifold, d≥2d\ge2. If its spectrum concentrates modulo one near ss fixed phases, with vanishing tolerance and o(Rd)o(R^d) exceptional eigenvalue copies along a sequence of cutoffs, then the time-2π2\pi Hamiltonian map has order at most ss, and there is a commuting quantizer AA. Finer spectral resolutions give sharp upper bounds on the order of P−AP-A. If P−AP-A is not smoothing, signed displacement profiles and Weyl mass determine the Liouville eigenvalue distribution of its normalized leading symbol.

For finite free isometric quotients of closed connected Zoll manifolds of dimension at least two, the Poisson relation is an equality, and the scalar spectrum determines the maximal dimension of returning components at each length. Among closed connected Riemannian nn-manifolds whose distinct scalar eigenvalues have multiplicities mj=cjn−1+O(jn−2)m_j=cj^{n-1}+O(j^{n-2}), the possible positive coefficients are exactly c=2n/((n−1)!N)c=2^n/((n-1)!N), N∈ℤ>0N\in\mathbb Z_{>0}. The largest value characterizes round real projective space up to scale.

MSC 2020

58J53Primary
Isospectrality
53C22Secondary
Geodesics in global differential geometry
58E10Secondary
Variational problems in applications to the theory of geodesics (problems in one independent variable)
55R20Secondary
Spectral sequences and homology of fiber spaces in algebraic topology
53D12Secondary
Lagrangian submanifolds; Maslov index

Mathematics Subject Classification 2020

Record

Version DOI
10.5281/zenodo.24153539
All versions
10.5281/zenodo.23307579
Subjects
Laplace spectrum; spherical space form; spectral rigidity; closed geodesics; Zoll manifold; Poisson relation
Language
English

Cite this paper

Anass Nifa. Spectral resolution and rigidity of spherical space forms. Preprint, Zenodo, v3 (2026). doi:10.5281/zenodo.24153539

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References

As listed in the manuscript bibliography.

  1. [1]

    A. Abbondandolo and G. Benedetti, On the local systolic optimality of Zoll contact forms, Geom. Funct. Anal. 33 (2023), no. 2, 299–363. doi:10.1007/s00039-023-00624-z. https://arxiv.org/abs/1912.04187.

    DOI
  2. [2]

    A. Abbondandolo, G. Benedetti and L. Polterovich, Lorentz–Finsler metrics on symplectic and contact transformation groups, Ann. Sc. Norm. Super. Pisa Cl. Sci., published online (2025). DOI: 10.2422/2036-2145.202305_003. Also arXiv:2210.02387v1.

    DOI · arXiv v1
  3. [3]

    J. Adams, Character of the oscillator representation, Israel J. Math. 98 (1997), 229–252. doi:10.1007/BF02937336. Author’s version: https://www.math.utah.edu/~ptrapa/math-library/adams/adams-char-of-oscillator.pdf.

    DOI
  4. [4]

    A. A. Agrachev, A. Gentile, and A. Lerario, Geodesics and horizontal-path spaces in Carnot groups, Geom. Topol. 19 (2015), no. 3, 1569–1630. doi:10.2140/gt.2015.19.1569.

    DOI
  5. [5]

    M. Amann, C. Lange and M. Radeschi, Odd-dimensional orbifolds with all geodesics closed are covered by manifolds, Math. Ann. 380 (2021), 1355–1386. DOI: 10.1007/s00208-021-02214-6.

    DOI
  6. [6]

    B. Ammann, R. Lauter, V. Nistor, and A. Vasy, Complex powers and non-compact manifolds, Comm. Partial Differential Equations 29 (2004), 671–705, doi:10.1081/PDE-120037329. https://arxiv.org/abs/math/0211305v1.

    DOI
  7. [7]

    S. Bates and A. Weinstein, Lectures on the Geometry of Quantization, Berkeley Mathematics Lecture Notes, vol. 8, American Mathematical Society, Providence, RI, 1997. Author’s version: https://math.berkeley.edu/~alanw/GofQ.pdf.

  8. [8]

    V. N. Berestovskiĭ and Yu. G. Nikonorov, Clifford–Wolf homogeneous Riemannian manifolds, J. Differential Geom. 82 (2009), no. 3, 467–500. https://archive.ymsc.tsinghua.edu.cn/pacm_download/21/781-JDG-82-3-A1-berestovskii.pdf.

  9. [9]

    R. Bott, On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math. 9 (1956), 171–206. doi:10.1002/cpa.3160090204.

    DOI
  10. [10]

    G. E. Bredon, Topology and Geometry, Graduate Texts in Mathematics, vol. 139, Springer-Verlag, New York, 1993. doi:10.1007/978-1-4757-6848-0.

    DOI
  11. [11]

    K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York–Berlin, 1982. doi:10.1007/978-1-4684-9327-6.

    DOI
  12. [12]

    A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer-Verlag, Berlin, 2001. DOI: 10.1007/978-3-540-45330-7. Author’s version, revised January 2006.

    DOI
  13. [13]

    H. H. Chan, S. H. Chan, and Z.-G. Liu, Elementary derivations of the Rogers–Fine identity and other q-series identities, Discrete Math. 348 (2025), 114387. doi:10.1016/j.disc.2024.114387.

    DOI
  14. [14]

    D. Cianci, On the Poisson relation for flat and spherical space forms, arXiv:1608.02816 (2016). https://arxiv.org/abs/1608.02816.

  15. [15]

    Y. Colin de Verdière, Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes périodiques, Comment. Math. Helv. 54 (1979), 508–522. doi:10.1007/BF02566290.

    DOI
  16. [16]

    Y. Colin de Verdière, Spectrum of the Laplace operator and periodic geodesics: thirty years after, Ann. Inst. Fourier 57 (2007), 2429–2463, doi:10.5802/aif.2339. https://aif.centre-mersenne.org/item/10.5802/aif.2339.pdf.

    DOI
  17. [17]

    M. Colantonio and E. A. Lauret, The fundamental group of a spherical space form is not audible, Doc. Math. (2026), published online first. https://doi.org/10.4171/DM/1102.

  18. [18]

    F. Dai and Y. Xu, Spherical harmonics, arXiv:1304.2585v1 (2013), Chapter 1 of Approximation Theory and Harmonic Analysis on Spheres and Balls. https://arxiv.org/abs/1304.2585v1.

  19. [19]

    S. Deng and M. Xu, Clifford–Wolf translations of Finsler spaces, Forum Math. 26 (2014), no. 5, 1413–1428. doi:10.1515/forum-2012-0032. https://arxiv.org/abs/1201.3714v1.

    DOI
  20. [20]

    G. de Rham, S. Maumary and M. A. Kervaire, Torsion et type simple d’homotopie, Lecture Notes in Mathematics, no. 48, Springer-Verlag, Berlin–Heidelberg–New York, 1967. https://webhomes.maths.ed.ac.uk/~v1ranick/papers/derkerma.pdf.

  21. [21]

    J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), 39–79. doi:10.1007/BF01405172.

    DOI
  22. [22]

    J. J. Duistermaat and L. Hörmander, Fourier integral operators. II, Acta Math. 128 (1972), 183–269. doi:10.1007/BF02392165.

    DOI
  23. [23]

    S. Dwivedi, J. Herman, L. C. Jeffrey, and T. van den Hurk, Hamiltonian Group Actions and Equivariant Cohomology, SpringerBriefs in Mathematics, Springer, Cham, 2019. doi:10.1007/978-3-030-27227-2.

    DOI
  24. [24]

    P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, and E. Yudovina, Introduction to Representation Theory, Student Mathematical Library, vol. 59, American Mathematical Society, Providence, RI, 2011. doi:10.1090/stml/059. https://math.mit.edu/~etingof/representationtheorybook.pdf.

    DOI
  25. [25]

    V. Fischer, Real Trace Expansions, Doc. Math. 24 (2019), 2159–2202. doi:10.4171/DM/723.

    DOI
  26. [26]

    W. Franz, Über die Torsion einer Überdeckung, J. reine angew. Math. 173 (1935), 245–254. doi:10.1515/crll.1935.173.245.

    DOI
  27. [27]

    C. Garetto, C. Jäh, and M. Ruzhansky, Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I: well-posedness, Math. Ann. 372 (2018), 1597–1629. doi:10.1007/s00208-018-1672-1.

    DOI
  28. [28]

    P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, Publish or Perish, 1984; electronic reprint, 1996, https://pages.uoregon.edu/gilkey/dirPDF/InvarianceTheory1Ed.pdf.

  29. [29]

    C. Gordon, W. Kirwin, D. Schueth and D. Webb, Quantum equivalent magnetic fields that are not classically equivalent, Ann. Inst. Fourier 60 (2010), no. 7, 2403–2419. doi:10.5802/aif.2612.

    DOI
  30. [30]

    K. Grove, Condition (C) for the energy integral on certain path spaces and applications to the theory of geodesics, J. Differential Geom. 8 (1973), no. 2, 207–223. doi:10.4310/jdg/1214431639.

    DOI
  31. [31]

    K. Grove and M. Tanaka, On the number of invariant closed geodesics, Acta Math. 140 (1978), 33–48. doi:10.1007/BF02392302.

    DOI
  32. [32]

    V. Guillemin and K. Okikiolu, Szegő theorems for Zoll operators, Math. Res. Lett. 3 (1996), 449–452.

  33. [33]

    V. Guillemin and K. Okikiolu, Spectral asymptotics of Toeplitz operators on Zoll manifolds, J. Funct. Anal. 146 (1997), no. 2, 496–516. doi:10.1006/jfan.1996.3044.

    DOI
  34. [34]

    V. Guillemin, A. Uribe, and Z. Wang, Semiclassical states associated with isotropic submanifolds of phase space, Lett. Math. Phys. 106 (2016), no. 12, 1695–1728. doi:10.1007/s11005-016-0853-7. arXiv:1511.01241v2.

    DOI · arXiv v2
  35. [35]

    J. Gutt, Generalized Conley–Zehnder index, Ann. Fac. Sci. Toulouse Math. (6) 23 (2014), 907–932. arXiv:1307.7239.

    arXiv
  36. [36]

    J. Gutt, Normal forms for symplectic matrices, Portugaliae Math. (N.S.) 71 (2014), no. 2, 109–139. doi:10.4171/PM/1944. https://ems.press/content/serial-article-files/44732.

    DOI
  37. [37]

    A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.

  38. [38]

    A. Hatcher, Vector Bundles and K-Theory, version 2.2, November 2017. https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf.

  39. [39]

    J. Hedicke, A causal characterization of Sp^+_ell(2n), Commun. Contemp. Math. 26 (2024), no. 7, 2350040, 13 pp. DOI: 10.1142/S0219199723500402. arXiv:2211.03707v1.

    DOI · arXiv v1
  40. [40]

    J. W. Helton, An operator algebra approach to partial differential equations; propagation of singularities and spectral theory, Indiana Univ. Math. J. 26 (1977), no. 6, 997–1018. doi:10.1512/iumj.1977.26.26081. https://mathweb.ucsd.edu/~helton/BILLSPAPERSscanned/H77a.pdf.

    DOI
  41. [41]

    M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York–Heidelberg, 1976. https://math.stanford.edu/~ralph/math215b/Hirsch.pdf.

  42. [42]

    L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218. doi:10.1007/BF02391913.

    DOI
  43. [43]

    W.-Y. Hsiang, Cohomology Theory of Topological Transformation Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 85, Springer-Verlag, Berlin–Heidelberg, 1975. doi:10.1007/978-3-642-66052-8.

    DOI
  44. [44]

    E. Humbert, Y. Privat, and E. Trélat, Geometric and spectral characterization of Zoll manifolds, invariant measures and quantum limits, arXiv:1811.12717 (2018). https://arxiv.org/abs/1811.12717.

  45. [45]

    A. Ikeda, On the spectrum of a Riemannian manifold of positive constant curvature, Osaka J. Math. 17 (1980), 75–93. DOI: 10.18910/5171.

    DOI
  46. [46]

    A. Ikeda and Y. Yamamoto, On the spectra of 3-dimensional lens spaces, Osaka J. Math. 16 (1979), 447–469. doi:10.18910/4811.

    DOI
  47. [47]

    M. Izydorek and W. Marzantowicz, Equivariant maps between cohomology spheres, Topol. Methods Nonlinear Anal. 5 (1995), 279–289. doi:10.12775/TMNA.1995.019.

    DOI
  48. [48]

    D. Jakobson and A. Strohmaier, High energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows, Communications in Mathematical Physics 270 (2007), no. 3, 813–833. DOI: 10.1007/s00220-006-0176-0. https://arxiv.org/abs/math/0607616.

    DOI
  49. [49]

    T. Kato, Perturbation Theory for Linear Operators, corrected printing of the second edition, Grundlehren der mathematischen Wissenschaften, vol. 132, Springer-Verlag, Berlin–New York, 1980.

  50. [50]

    D. Kelmer, A refinement of strong multiplicity one for spectra of hyperbolic manifolds, Trans. Amer. Math. Soc. 366 (2014), no. 11, 5925–5961. doi:10.1090/S0002-9947-2014-06102-3. https://arxiv.org/abs/1108.2977v2.

    DOI
  51. [51]

    S. Kobayashi, Transformation Groups in Differential Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 70, Springer-Verlag, 1972; reprinted in Classics in Mathematics, 1995. doi:10.1007/978-3-642-61981-6.

    DOI
  52. [52]

    R. Kuwabara, Some spectral results for the Laplacian on line bundles over S^n, Comment. Math. Helv. 59 (1984), 439–458. doi:10.5169/seals-45404.

    DOI
  53. [53]

    R. Kuwabara, Spectrum of the Laplacian on vector bundles over C_{2π}-manifolds, J. Differential Geom. 27 (1988), no. 2, 241–258. doi:10.4310/jdg/1214441782.

    DOI
  54. [54]

    R. Kuwabara, Correction and addendum to: On the classical and the quantum mechanics in a magnetic field, J. Math. Tokushima Univ. 30 (1996), 81–87. https://www-math.st.tokushima-u.ac.jp/journal/1996/1996-6.pdf.

  55. [55]

    E. A. Lauret, A computational study on lens spaces isospectral on forms, Experimental Mathematics 30 (2021), no. 2, 268–282. doi:10.1080/10586458.2018.1538908. The theorem and remark numbers refer to arXiv:1703.03077v3 (9 May 2017), https://arxiv.org/abs/1703.03077v3.

    DOI
  56. [56]

    E. A. Lauret and R. J. Miatello, Strong multiplicity one theorems for locally homogeneous spaces of compact-type, Proc. Amer. Math. Soc. 148 (2020), no. 7, 3163–3173. doi:10.1090/proc/14980. https://arxiv.org/abs/1804.08288v2.

    DOI
  57. [57]

    E. A. Lauret, R. J. Miatello, and J. P. Rossetti, Spectra of lens spaces from 1-norm spectra of congruence lattices, Int. Math. Res. Not. IMRN 2016 (2016), no. 4, 1054–1089. doi:10.1093/imrn/rnv159. https://arxiv.org/abs/1311.7167v4.

    DOI
  58. [58]

    J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Graduate Texts in Mathematics, vol. 176, Springer-Verlag, New York, 1997. doi:10.1007/b98852.

    DOI
  59. [59]

    S. Lin and B. Schmidt, Real projective spaces with all geodesics closed, Geom. Funct. Anal. 27 (2017), 631–636. DOI: 10.1007/s00039-017-0407-x.

    DOI
  60. [60]

    A. J. McNeil and J. Nešlehová, Multivariate Archimedean copulas, d-monotone functions and ℓ_1-norm symmetric distributions, Ann. Statist. 37 (2009), no. 5B, 3059–3097. doi:10.1214/07-AOS556. https://arxiv.org/abs/0908.3750v1.

    DOI
  61. [61]

    J. Milnor, Morse Theory, Annals of Mathematics Studies, no. 51, Princeton University Press, 1963.

  62. [62]

    E. Meinrenken, Group actions on manifolds, lecture notes, University of Toronto, Spring 2003. https://www.math.toronto.edu/mein/teaching/LectureNotes/action.pdf.

  63. [63]

    V. Ozols, Critical points of the displacement function of an isometry, J. Differential Geom. 3 (1969), no. 3–4, 411–432. doi:10.4310/jdg/1214429062.

    DOI
  64. [64]

    R. S. Palais and C.-L. Terng, Critical Point Theory and Submanifold Geometry, Lecture Notes in Mathematics, vol. 1353, Springer-Verlag, Berlin, 1988. doi:10.1007/BFb0087442.

    DOI
  65. [65]

    G. P. Paternain, Transparent connections over negatively curved surfaces, J. Mod. Dyn. 3 (2009), 311–333. doi:10.3934/jmd.2009.3.311. arXiv:0809.4360, version 4 (2010).

    DOI · arXiv
  66. [66]

    P. Petersen, Riemannian Geometry, second edition, Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006. doi:10.1007/978-0-387-29403-2.

    DOI
  67. [67]

    R. Ponge, Connes’ integration and Weyl’s laws, J. Noncommut. Geom. 17 (2023), 719–767, doi:10.4171/JNCG/509.

    DOI
  68. [68]

    F. Prüfer, On the spectrum and the geometry of the spherical space forms, I, Ann. Global Anal. Geom. 3 (1985), 129–154. doi:10.1007/BF01000336.

    DOI
  69. [69]

    F. Prüfer, On the spectrum and the geometry of a spherical space form II, Ann. Global Anal. Geom. 3 (1985), 289–312. doi:10.1007/BF00130482.

    DOI
  70. [70]

    M. Radeschi and B. Wilking, On the Berger conjecture for manifolds all of whose geodesics are closed, Invent. Math. 210 (2017), 911–962. DOI: 10.1007/s00222-017-0742-4. Also arXiv:1511.07852.

    DOI · arXiv
  71. [71]

    S. Ramanujan, On certain trigonometrical sums and their applications in the theory of numbers, Trans. Cambridge Philos. Soc. 22 (1918), no. 13, 259–276. https://ramanujan.sirinudi.org/Volumes/published/ram21.pdf.

  72. [72]

    J.-P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics, vol. 7, Springer-Verlag, New York, 1973. doi:10.1007/978-1-4684-9884-4.

    DOI
  73. [73]

    S. Smale, Generalized Poincaré’s conjecture in dimensions greater than four, Ann. of Math. (2) 74 (1961), 391–406.

  74. [74]

    R. P. Stanley, Enumerative Combinatorics, Vol. 1, second edition, Cambridge University Press, 2012. Author’s version of 15 July 2011, https://math.mit.edu/~rstan/ec/ec1.pdf.

  75. [75]

    E. M. Stein and R. Shakarchi, Fourier Analysis: An Introduction, Princeton Lectures in Analysis, vol. 1, Princeton University Press, Princeton, NJ, 2003.

  76. [76]

    C. J. Sutton, Detecting the moments of inertia of a molecule via its rotational spectrum, unpublished manuscript, 2013. https://math.dartmouth.edu/~cjsutton/Sutton.2013.Moments1.pdf.

  77. [77]

    C. J. Sutton, On the Poisson relation for compact Lie groups, Ann. Global Anal. Geom. 57 (2020), 537–589. DOI: 10.1007/s10455-020-09712-x. Also arXiv:1606.07426.

    DOI · arXiv
  78. [78]

    S. Tanno, Eigenvalues of the Laplacian of Riemannian manifolds, Tôhoku Math. J. (2) 25 (1973), 391–403. doi:10.2748/tmj/1178241341.

    DOI
  79. [79]

    M. E. Taylor, Short Course on Pseudodifferential Operators, author’s version, 2022. https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2022/05/psidolect.pdf.

  80. [80]

    G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, Graduate Studies in Mathematics, vol. 99, American Mathematical Society, Providence, RI, 2009. https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf.

  81. [81]

    C. T. C. Wall, Free actions of finite groups on spheres, in Algebraic and Geometric Topology, Part 1 (R. J. Milgram, ed.), Proc. Sympos. Pure Math., vol. 32, American Mathematical Society, Providence, RI, 1978, 115–124.

  82. [82]

    L. C. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York–Heidelberg–Berlin, 1982. doi:10.1007/978-1-4684-0133-2.

    DOI
  83. [83]

    A. Weinstein, On the volume of manifolds all of whose geodesics are closed, J. Differential Geom. 9 (1974), 513–517. doi:10.4310/jdg/1214432547.

    DOI
  84. [84]

    B. Wilking, Index parity of closed geodesics and rigidity of Hopf fibrations, Invent. Math. 144 (2001), 281–295. doi:10.1007/PL00005801.

    DOI
  85. [85]

    J. A. Wolf, Isospectrality for spherical space forms, Results Math. 40 (2001), 321–338. doi:10.1007/BF03322715.

    DOI
  86. [86]

    J.-Y. Wu, On almost isospectral manifolds, Indiana Univ. Math. J. 39 (1990), no. 4, 1373–1381. doi:10.1512/iumj.1990.39.39061.

    DOI
  87. [87]

    C. T. Yang, Odd-dimensional Wiedersehen manifolds are spheres, J. Differential Geom. 15 (1980), no. 1, 91–96. doi:10.4310/jdg/1214435386.

    DOI
  88. [88]

    S. Zelditch, Maximally degenerate Laplacians, Ann. Inst. Fourier (Grenoble) 46 (1996), 547–587. doi:10.5802/aif.1524.

    DOI
  89. [89]

    S. Zelditch, Index and dynamics of quantized contact transformations, Ann. Inst. Fourier 47 (1997), no. 1, 305–363. doi:10.5802/aif.1568.

    DOI
  90. [90]

    M. Egidi, K. Gittins, G. Habib and N. Peyerimhoff, Eigenvalue estimates for the magnetic Hodge Laplacian on differential forms, J. Spectr. Theory 13 (2023), 1297–1343. doi:10.4171/JST/480.

    DOI
  91. [91]

    D.-W. Chiou, D.-H. Lee and W.-Y. Hsiang, Eigen wavefunctions of a charged particle moving in a self-linking magnetic field, arXiv:math-ph/0411044v1 (2004). https://arxiv.org/abs/math-ph/0411044v1.