ABSTRACT (original article): Motivated by Nirenberg’s problem on isometric rigidity of tight surfaces, we study closed asymptotic curves
Γ
on negatively curved surfaces
M
in Euclidean 3-space. In particular, using Călugăreanu’s theorem, we obtain a formula for the linking number
Lk(Γ,n)
of Γ with the normal n of
M
. It follows that when
Lk(Γ,n)=0
,
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
Lk(Γ,n)≠0
, and discuss various restrictions on
Γ
when
n
is injective. CITATION (original article): Ghomi, M., Raffaelli, M. Topology of closed asymptotic curves on negatively curved surfaces. J Geom Anal 35, 381 (2025). https://doi.org/10.1007/s12220-025-02200-3
This notebook is an accompaniment to the Section 4 of the paper Topology of closed asymptotic curves on negatively curved surfaces.
Note: Due to conflicting functions in the sections below, Mathematica Kernel needs to be restarted when running one section after another.
First Example
We start with the example given by Kovaleva.

Definition

In[]:=
r=Sqrt[63/8];a=1/8;
In[]:=
c[t_]={(3+Sin[t])Cos[rCos[t]],(3+Sin[t])Sin[rCos[t]],a(Sin[2t]+46Cos[t]-27Cos[t]^3+(27/5)Cos[t]^5)};
In[]:=
ParametricPlot3D[c[t],{t,0,2Pi}]
Out[]=
In[]:=
ParametricPlot[{c[t][[1]],c[t][[2]]},{t,0,2Pi}]
Out[]=
-3
-2
-1
1
2
3
4
-4
-2
2
4

Spherical Images

In[]:=
T[t_]=c'[t]/Sqrt[c'[t].c'[t]];
In[]:=
g=ParametricPlot3D[T[t],{t,0,2Pi}];​​Show[{Graphics3D[{Opacity[0.6],Sphere[]}],Graphics3D[Point[{0,0,1}]],g},BoxedFalse]
Out[]=
The principal normal is given by
In[]:=
PN[t_]=T'[t]/Sqrt[T'[t].T'[t]];
In[]:=
Plot[T'[t].T'[t],{t,0,2Pi}]
Out[]=
1
2
3
4
5
6
2
4
6
8
10
12
In[]:=
FindRoot[T'[t].T'[t],{t,3}]
Out[]=
{t3.48143}
In[]:=
FindRoot[T'[t].T'[t],{t,6}]
Out[]=
{t5.94335}
So we may define the vector
⟂
n
as:
In[]:=
nperp[t_]=If[3.48143<t<5.94335,PN[t],-PN[t]];
In[]:=
g=ParametricPlot3D[nperp[t],{t,0,2Pi}];​​Show[Graphics3D[{Opacity[0.6],Sphere[]}],g,BoxedFalse]
Out[]=
In[]:=
n[t_]=Cross[T[t],nperp[t]];
In[]:=
g=ParametricPlot3D[n[t],{t,0,2Pi}];​​Show[{Graphics3D[{RGBColor[1,1,1,0.25],Sphere[]}],g},BoxedFalse]
Out[]=

Curvatures and Geodesic Torsion

Composite Picture

Second Example
Here we construct a simple closed curve Γ with an injective Darboux binormal n such that Lk(Γ,n)=2.

The Binormal

The Tangent

The Function ρ

The Curve

Curvature and Geodesic Torsion

So there are two inflections.
So the torsion is indeed nonvanishing.

Composite Picture

CITE THIS NOTEBOOK

Topology of closed asymptotic curves on negatively curved surfaces​
by Mohammad Ghomi & Matteo Raffaelli
Wolfram Community, STAFF PICKS, January 13, 2025
​https://community.wolfram.com/groups/-/m/t/3356757