Journal of Mathematical Physics, Analysis, Geometry
2020, vol. 16, No 3, pp. 312-363   https://doi.org/10.15407/mag16.03.312     ( to contents , go back )
https://doi.org/10.15407/mag16.03.312

On the Cauchy-Riemann Geometry of Transversal Curves in the 3-Sphere

Emilio Musso

Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
E-mail: emilio.musso@polito.it

Lorenzo Nicolodi

Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze 53/A, I-43124 Parma, Italy
E-mail: lorenzo.nicolodi@unipr.it

Filippo Salis

Istituto Nazionale di Alta Matematica, Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
E-mail: filippo.salis@gmail.com

Received March 18, 2020.

Abstract

Let $\mathrm S^3$ be the unit sphere of $\mathbb C^2$ with its standard Cauchy--Riemann (CR) structure. This paper investigates the CR geometry of curves in $\mathrm S^3$ which are transversal to the contact distribution, using the local CR invariants of $\mathrm S^3$. More specifically, the focus is on the CR geometry of transversal knots. Four global invariants of transversal knots are considered: the phase anomaly, the CR spin, the Maslov index, and the CR self-linking number. The interplay between these invariants and the Bennequin number of a knot are discussed. Next, the simplest CR invariant variational problem for generic transversal curves is considered and its closed critical curves are studied.

Mathematics Subject Classification 2000: 53C50, 53C42, 53A10
Key words: CR geometry of the 3-sphere, contact geometry, transversal curves, CR invariants of transversal knots, self-linking number, Bennequin number, the strain functional for transversal curves, critical knots

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