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

SIGMA 19 (2023), 101, 36 pages      arXiv:2307.04763      https://doi.org/10.3842/SIGMA.2023.101
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

Abstract
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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