Page 391 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry


Notes We can define the orthonormal frame (e , e , N), known as the Darboux frame, where e and


1
1
2

2 e are unit vectors corresponding to the principal directions, N is the normal to the surface at
 
X(u(0), v(0)), and N = e × e .
2
1
dN
It is interesting to study the quantity (u,v) (0).
ds

If t = C’(0) is the unit tangent vector at X(u(0), v(0)), we have another orthonormal frame
   
considered in previous Section, namely (t, n , N), where n = N × t , and if  is the angle
g
g
 
between e and t we have
1
  
t = cos  1 e + sin  2 e ,
  
g n = –sin e + cos 1  2 e .
Lemma 8. Given a curve C : s  X(u(s), v(s)) parameterized by arc length on a surface X, we have
dN (u,v) k t T 

ds (0)   N  g g n ,
where k is the normal curvature, and where the geodesic torsion T is given by
N
g
T = (k – k ) sin  cos .
1
g
2
From the formula
T = (k – k ) sin  cos ,
g
1
2
since  is the angle between the tangent vector to the curve C and a principal direction, it is clear
that the lines of curvatures are characterized by the fact that T = 0.
g
One will also observe that orthogonal curves have opposite geodesic torsions (same absolute
value and opposite signs).

If n is the principal normal, T is the torsion of C at X(u(0), v(0)), and  is the angle between N and
 
.
n so that cos  = N n , we claim that
T = T – d ,
ds
g
which is often known as Bonnet’s formula.
Lemma 9. Given a curve C : s  X(u(s), v(s)) parameterized by arc length on a surface X, the
geodesic torsion T is given by
g
T = T – d = (k – k ) sin  cos ,
ds
2
1
g
where T is the torsion of C at X(u(0), v(0)), and is the angle between N and the principal normal

n to C at s = 0.

Notes The geodesic torsion only depends on the tangent of curves C. Also, for a curve
for which  = 0, we have T = T.
g

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