Topology of closed asymptotic curves on negatively curved surfaces

Abstract Motivated by Nirenberg’s problem on isometric rigidity of tight surfaces, we study closed asymptotic curves $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> on negatively curved surfaces M in Euclidean 3-space. In particular, using Călugăreanu’s theorem, we obtain a formula for the linking number $$\text {Lk}(\Gamma ,n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>Lk</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>Γ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> with the normal n of M . It follows that when $$\text {Lk}(\Gamma , n)=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>Lk</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>Γ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> 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 $$\text {Lk}(\Gamma , n)\ne 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>Lk</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>Γ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , and discuss various restrictions on $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> when n is injective.