Page 267 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes A regular curve C is parametrized by the arc-length parameter s, that is, c : L  En (L  s  c(s)
dc dc(s)
 En)([1]). Then the tangent vector field along C has unit length, that is,  1 for all
ds ds
s  L.
Hereafter, curves considered are regular C -curves in E parametrized by the arc-length parameter.
n

dc(s)
Let C be a curve in E , that is, c(s)  E for all s  L. Let t(s) = for all s  L. The vector field
n
n
ds
t is called a unit tangent vector field along C, and we assume that the curve C satisfies the
following conditions (C ) ~ (C ):
n–1
1
dt(s) d c(s)
2
(C ) : k (s)  ds  ds 2  0 for all s L.

1
1
Then we obtain a well-defined vector field n along C, that is, for all s  L,
1
1 dt(s)
n (s)  k1(s) . ds ,
1
and we obtain,
t(s),n (s)  0, n (s),n (s)  1.
1
1
1
dn (s)
1
(C ) : k (s)  ds  k (s).t(s)  0 for all s L.

2
1
2
Then we obtain a well-defined vector field n along C, that is, for all s  L,
2
1  dn (s) 
n (s)  .  1  k (s).t(s) ,

2
1
k (s)  ds 
2
and we obtain, for i, j = 1, 2,
t(s),n (s)  0, n (s),n (s)   ij ,
i
j
i
where  denotes the Kronecker’s symbol.
ij
By an inductive procedure, for  = 3, 4, ..., n – 2,
dn (s)


(C ) : k (s)   ds 1  k   1 (s).n   2 (s)  0 for all s L.


Then we obtain, for  = 3, 4, ..., n – 2, a well-defined vector field n along C, that is, for all s  L

1  dn (s) 
n (s)  .    1  k   1 (s).n   2 (s) ,


k (s)  ds 

and for i, j = 1, 2, ..., n – 2
t(s),n (s)  0, n (s),n (s)   ij .
j
i
i

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