Page 258 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 21: New Spherical Indicatrices and their Characterizations

Similar to the M Bishop spherical image, one can have Notes
1
ds
T = T and    k . (9)
 ds 2
So, by differentiating of the formula (9), we get

ds

'
T  T ds   k M  k M 2
1
2
1

or, in another words,
k
'
T   k 1 2 M  M ,
2
1

since, we express
2
 k 

k  T  1   1  (10)
y

 k 2 
and
k M
N   k K 1  M  K  2 .
1

2
The cross product T × N gives us
 
1 k
B   K  M  k K 1  M .
1
2

2
By the formula of the torsion, we have
'
k 2   k k  
1
T   2  . (11)
2

k  k 2 2
1
In terms of equations (10) and (11) and by the Theorem 2, we may obtain:
Corollary 3. Let  = (s ) be the M spherical image of a regular curve  = (s). If  = (s) is a
2

B-slant helix, then the M Bishop spherical image (s ) is a circle in the osculating plane.
2

Theorem 6. Let  = (s ) be the M spherical image of a regular curve  = (s). Then, there exists
2

a relation among Frenet-Serret invariants of (s ) and the Bishop curvatures of  = (s) as

follows:
k 1 s  2 0.
k 2   0 k T ds 



Proof. Similar to proof of the theorem 6, above equation can be obtained by the equations (9),
(10) and (11).
In the light of the propositions 4 and 5, we also give the following theorems for the curve
 = (s ):


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