Topology of closed asymptotic curves on negatively curved surfaces

TL;DR

This paper investigates the topology of closed asymptotic curves on negatively curved surfaces in 3D space, deriving linking number formulas and exploring geometric restrictions, extending previous observations and providing new examples.

Contribution

It introduces a linking number formula for asymptotic curves on negatively curved surfaces and analyzes the implications for their geometric projections and properties.

Findings

Linking number formula for asymptotic curves and surface normals.

Curves with zero linking number cannot have certain planar projections.

Constructed example with injective normal but nonzero linking number.

Abstract

Motivated by Nirenberg's problem on isometric rigidity of tight surfaces, we study closed asymptotic curves $\Gamma$ on negatively curved surfaces $M$ in Euclidean $3$-space. In particular, using C\u{a}lug\u{a}reanu's theorem, we obtain a formula for the linking number $Lk(\Gamma,n)$ of $\Gamma$ with the normal $n$ of $M$. It follows that when $Lk(\Gamma, n)=0$, $\Gamma$ cannot have any locally star-shaped planar projections with vanishing crossing number, which extends observations of Kovaleva, Panov and Arnold. These results hold also for curves with nonvanishing torsion and their binormal vector field. Furthermore we construct an example where $n$ is injective but $Lk(\Gamma, n)\neq 0$, and discuss various restrictions on $\Gamma$ when $n$ is injective.