Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 101, 36 pages      arXiv:2307.04763
Contribution to the Special Issue on Symmetry, Invariants, and their Applications in honor of Peter J. Olver

On the Total CR Twist of Transversal Curves in the 3-Sphere

Emilio Musso a and Lorenzo Nicolodi b
a) Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
b) Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze 53/A, I-43124 Parma, Italy

Received July 11, 2023, in final form November 26, 2023; Published online December 21, 2023

We investigate the total CR twist functional on transversal curves in the standard CR 3-sphere $\mathrm S^3 \subset \mathbb C^2$. The question of the integration by quadratures of the critical curves and the problem of existence and properties of closed critical curves are addressed. A procedure for the explicit integration of general critical curves is provided and a characterization of closed curves within a specific class of general critical curves is given. Experimental evidence of the existence of infinite countably many closed critical curves is provided.

Key words: CR 3-sphere; transversal curves; CR invariants; total CR twist; Griffiths' formalism; Lax formulation of E-L equations; integration by quadratures; closed critical curves.

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  1. Bennequin D., Entrelacements et équations de Pfaff, Astérisque 107 (1983), 87-161.
  2. Bryant R.L., On notions of equivalence of variational problems with one independent variable, in Differential Geometry: the Interface Between Pure and Applied Mathematics (San Antonio, Tex., 1986), Contemp. Math., Vol. 68, American Mathematical Society, Providence, RI, 1987, 65-76.
  3. Calabi E., Olver P.J., Shakiban C., Tannenbaum A., Haker S., Differential and numerically invariant signature curves applied to object recognition, Int. J. Comput. Vis. 26 (1998), 107-135.
  4. Calini A., Ivey T., Integrable geometric flows for curves in pseudoconformal $S^3$, J. Geom. Phys. 166 (2021), 104249, 17 pages.
  5. Cartan E., Sur la géométrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes II, Ann. Scuola Norm. Super. Pisa Cl. Sci. (2) 1 (1932), 333-354.
  6. Cartan E., Sur la géométrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes, Ann. Mat. Pura Appl. 11 (1933), 17-90.
  7. Chern S.S., Moser J.K., Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219-271.
  8. Chiu H.-L., Ho P.T., Global differential geometry of curves in three-dimensional Heisenberg group and CR sphere, J. Geom. Anal. 29 (2019), 3438-3469.
  9. Dzhalilov A., Musso E., Nicolodi L., Conformal geometry of timelike curves in the $(1+2)$-Einstein universe, Nonlinear Anal. 143 (2016), 224-255, arXiv:1603.01035.
  10. Eliashberg Y., Legendrian and transversal knots in tight contact $3$-manifolds, in Topological Methods in Modern Mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, 171-193.
  11. Eshkobilov O., Musso E., Nicolodi L., The geometry of conformal timelike geodesics in the Einstein universe, J. Math. Anal. Appl. 495 (2021), 124730, 32 pages.
  12. Etnyre J.B., Introductory lectures on contact geometry,in Topology and Geometry of Manifolds (Athens, GA, 2001), Proc. Sympos. Pure Math., Vol. 71, American Mathematical Society, Providence, RI, 2003, 81-107, arXiv:math.SG/0111118.
  13. Etnyre J.B., Transversal torus knots, Geom. Topol. 3 (1999), 253-268, arXiv:math.GT/9906195.
  14. Etnyre J.B., Legendrian and transversal knots, in Handbook of Knot Theory, Elsevier, Amsterdam, 2005, 105-185, arXiv:math.SG/0306256.
  15. Etnyre J.B., Honda K., Knots and contact geometry I: Torus knots and the figure eight knot, J. Symplectic Geom. 1 (2001), 63-120, arXiv:math.GT/0006112.
  16. Fels M., Olver P.J., Moving coframes: I. A practical algorithm, Acta Appl. Math. 51 (1998), 161-213.
  17. Fels M., Olver P.J., Moving coframes: II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), 127-208.
  18. Fuchs D., Tabachnikov S., Invariants of Legendrian and transverse knots in the standard contact space, Topology 36 (1997), 1025-1053.
  19. Griffiths P.A., Exterior differential systems and the calculus of variations, Progr. Math., Vol. 25, Birkhäuser, Boston, MA, 1983.
  20. Hoffman W.C., The visual cortex is a contact bundle, Appl. Math. Comput. 32 (1989), 137-167.
  21. Hsu L., Calculus of variations via the Griffiths formalism, J. Differential Geom. 36 (1992), 551-589.
  22. Jacobowitz H., Chains in CR geometry, J. Differential Geom. 21 (1985), 163-194.
  23. Klein F., Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, Birkhäuser, Basel, 1993.
  24. Kogan I.A., Ruddy M., Vinzant C., Differential signatures of algebraic curves, SIAM J. Appl. Algebra Geom. 4 (2020), 185-226, arXiv:1812.11388.
  25. Musso E., Liouville integrability of a variational problem for Legendrian curves in the three-dimensional sphere, in Selected Topics in Cauchy-Riemann Geometry, Quad. Mat., Vol. 9, Seconda Università di Napoli, Caserta, 2001, 281-306.
  26. Musso E., Grant J.D.E., Coisotropic variational problems, J. Geom. Phys. 50 (2004), 303-338, arXiv:math.DG/0307216.
  27. Musso E., Nicolodi L., Closed trajectories of a particle model on null curves in anti-de Sitter 3-space, Classical Quantum Gravity 24 (2007), 5401-5411, arXiv:0709.2017.
  28. Musso E., Nicolodi L., Reduction for constrained variational problems on 3-dimensional null curves, SIAM J. Control Optim. 47 (2008), 1399-1414.
  29. Musso E., Nicolodi L., Invariant signatures of closed planar curves, J. Math. Imaging Vision 35 (2009), 68-85.
  30. Musso E., Nicolodi L., Quantization of the conformal arclength functional on space curves, Comm. Anal. Geom. 25 (2017), 209-242, arXiv:1501.04101.
  31. Musso E., Nicolodi L., Salis F., On the Cauchy-Riemann geometry of transversal curves in the 3-sphere, J. Math. Phys. Anal. Geom. 16 (2020), 312-363, arXiv:2004.11350.
  32. Musso E., Salis F., The Cauchy-Riemann strain functional for Legendrian curves in the 3-sphere, Ann. Mat. Pura Appl. 199 (2020), 2395-2434, arXiv:2003.01713.
  33. Nash O., On Klein's icosahedral solution of the quintic, Expo. Math. 32 (2014), 99-120, arXiv:1308.0955.
  34. Olver P.J., Applications of Lie groups to differential equations, Grad. Texts in Math., Vol. 107, Springer, New York, 1993.
  35. Olver P.J., Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995.
  36. Petitot J., Elements of neurogeometry, Lect. Notes Morphog., Springer, Cham, 2017.
  37. Storn R., Price K., Differential evolution - A simple and efficient heuristic for global optimization over continuous spaces, J. Global Optim. 11 (1997), 341-359.
  38. Trott M., Adamchik V., Solving the quintic with textscMathematica, available at

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