Page 253 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes The Bishop frame or parallel transport frame is an alternative approach to defining a moving
frame that is well defined even when the curve has vanishing second derivative. One can express
parallel transport of an orthonormal frame along a curve simply by parallel transporting each
component. The tangent vector and any convenient arbitrary basis for the remainder of the
frame are used.

 T'   0 k 1 k 2   T 
 M '   k   M .

 1    1 0 0   1  (1)

'
 M   k   M 
 2   2 0 0   2 
Here, we shall call the set {T, M , M } as Bishop trihedra and k and k as Bishop curvatures. The
1
2
1
2
relation matrix may be expressed as
 T  1  0 0   T 
   cos (s) sin (s)  



N 
   0   M 1  ,
   0  sin (s) cos (s)  M 

B

     2 
k
where (s) = arctan 2 , T(s) = ’(s) and k(s) = k  k . Here, Bishop curvatures are defined by
2
2
k 1 1 2
 k = K cos µ(s)
1
 k = K sin µ(s) .
 2
Izumiya and Takeuchi have introduced the concept of slant helix in the Euclidean 3-space E 3
saying that the normal lines makes a constant angle with a fixed direction. They characterized a
slant helix by the condition that the function
k 2  T  '
2 3 /2 
2
(k  t )  K  
is constant. In further researches, spherical images, the tangent and the binormal indicatrix and
some characterizations of such curves are presented. In the same space, the authors defined and
gave some characterizations of slant helices according to Bishop frame with the following
definition and theorem:
Definition 1. A regular curve  : I  E is called a slant helix according to Bishop frame provided
3
the unit vector M (s) of  has constant angle  with some fixed unit vector u; that is,
1
M , u = cos 
1
for all s  I.
Theorem 1. Let  : I  E be a unit speed curve with nonzero natural curvatures. Then  is a slant
3
helix if and only if

k 1 constant.
k 2 

To separate a slant helix according to Bishop frame from that of Frenet-Serret frame, in the rest
of the paper, we shall use notation for the curve defined above as “B-slant helix”.

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