Page 273 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes for all s  L. From the above fact, it holds

dcos (s) dsin (s)


ds  0, ds  0,
that is,  is a constant function on L with value  . Let  = (cos  ) (sin  ) be a constant number.
–1
0
0
0
Then (10.1) and (10.2) imply
k (s) – k (s) = k (s) ( s L),


3
2
1
that is, we obtain the relation (c).
Moreover, we obtain




 '(s)k ( (s)) . t( (s))   '(s)k ( (s)) . n ( (s))
2
2
1
= k (s)(cos (s) . t(s) {k (s)(cos (s)) k (s)(sin (s))} . n (s)






3
2
2
1
By the above equality and (3), we obtain


 '(s)k ( (s)) . n ( (s))   '(s)k ( (s)) . t( (s))


2
1
2
 k (s)(cos 0 ) . t(s) {k (s)(cos 0 ) k (s)(sin 0 )} . n (s)


2
2
3
1
= ( '(s)) {k ( (s))} . {A(s) . t(s) B(s) . n (s)},  2 1   1  2
where


A(s) = { '(s)k ( (s))} (1 1  2   k (s)) k (s)( k (s)   k (s))( k (s) k (s))


1
2
3
1
2
1



B(s) = { '(s)k ( (s))} ( k (s)) 1  2  2   k (s)) ( k (s)   k (s))( k (s) k (s))k (s)

2
1
2
2
3
3
 ( k (s)   k (s))(k (s)) 2

3
2
3
for all s  L. From (b) and (8), A(s) and B(s) are rewritten as:
2
2
2
2
2
1

A(s) = (  1) ( k (s)   k (s)) [(  1)k (s)k (s)   {(k (s))  (k (s))  (k (s)) }]



2
2
3
1
2
1
3
1
2
2
2
2
2


B(s) = (  1) ( k (s)   k (s)) [(  1)k (s)k (s)   {(k (s))  (k (s))  (k (s)) }].


2
3
2
3
1
1
2
Since '(s)k ( (s)) . n ( (s)) 2  2   0 for all s  L, it holds
2
2
2
2
(  1)k (s)k (s)   {(k (s))  (k (s))  (k (s)) }  0
3
1
2
1
2
for all s  L. Thus, we obtain the relation (d).
(ii) We assume that C (c : L  E ) is a C -special Frenet curve in E with curvature functions k , k 2
4
4

1
and k satisfying the relation (a), (b), (c) and (d) for constant numbers a, b, g and d. Then we
3
define a C -curve C by

c(s) = c(s) +  . n (s) +  . n (s) (11)
3
1
for all s  L, where s is the arc-length parameter of C. Differentiating (11) with respect to s and
using the Frenet equations, we obtain
dc(s) k (s)) . n (s)


1
2
ds  (1   k (s)) . t(s) ( k (s)   3 2
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