Year: 2025
Type: article
Abstract: Motivated by Nirenberg’s problem on isometric rigidity of tight surfaces, we study closed asymptotic curves $$\Gamma $$ Γ on negatively curved surfaces M in Euclidean 3-space. In particular, using Călugăreanu’s theorem, we obtain a formula for the linking number $$\text {Lk}(\Gamma ,n)$$ Lk ( Γ , n ) of $$\Gamma $$ Γ with the normal n of M . It follows that when $$\text {Lk}(\Gamma , n)=0$$ Lk ( Γ , n ) = 0 , $$\Gamma $$ Γ 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 $$\text {Lk}(\Gamma , n)\ne 0$$ Lk ( Γ , n ) ≠ 0 , and discuss various restrictions on $$\Gamma $$ Γ when n is injective.
Source: Journal of Geometric Analysis
Authors: Mohammad Ghomi, Matteo Raffaelli
Institution: Georgia Institute of Technology
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
Funder: National Science Foundation
OpenAlex ID: W4415011028