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

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.

44 pages, 7 figures. This is an extended version of a talk presented at the Conference "Geometry, Differential Equations and Analysis" in memory of A. V. Pogorelov for his 100th birthday anniversary, June 17-21, 2019, Kharkiv, Ukraine