On the Cauchy–Riemann geometry of transversal curves in the 3-sphere

Emilio Musso   ; Lorenzo Nicolodi   ; Filippo Salis

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 16 (2020) no. 3, pp. 312-363

Résumé

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.

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