Page 279 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 22.5 Summary

Theorem A. If n  4, then no C -special Frenet curve in E is a Bertrand curve.
n


Let E be an n-dimensional Euclidean space with Cartesian coordinates (x , x ,..., x ). By a
2
n
1
n

parametrized curve C of class C , we mean a mapping c of a certain interval I into E given
n

by
1
 x (t)
 x (t) 
2
c(t)     t I.

 : 
 n 
  x (t) 
1
If dc(t)  dc(t) dc(t) 2  0 for all t  I, then C is called a regular curve in E . Here .,.
,
n
dt dt dt
denotes the Euclidean inner product on E . We refer to[2] for the details of curves in E .
n
n
In the case of Euclidean 3-space, the Frenet 1-normal vector fields n1 is already called the

principal normal vector field along C, and the Frenet 1-normal line is already called the
principal normal line of C at c(s).
A C -special Frenet curve C in E (c : L  E ) is called a Bertrand curve if there exist a C -
n
n



n

special Frenet curve C (c : L  E ), distinct from C, and a regular C -map  : L 
d (s)

L (s   (s),  0 for all s  L) such that curves C and C have the same 1-normal line
ds

at each pair of corresponding points c(s) and c(s)  c( (s)) under . Here s and s arc-
length parameters of C and C respectively. In this case, C is called a Bertrand mate of C.
The following results are well-known:
Theorem (the case of n = 2). Every C -plane curve is a Bertrand curve.

Theorem (the case of n = 3). A C -special Frenet curve in E with 1-curvature function
3

k and 2-curvature function k is a Bertrand curve if and only if there exists a linear relation
2
1
ak (s) + bk (s) = 1
2
1
for all s  L, where a and b are nonzero constant real numbers.
Let C and C be C -special Frenet curves in E and  : L  L a regular C -map such that
4




each point c(s) of C corresponds to the point c(s)  c( (s)) of C for s  L. Here s and s arc-
length parameters of C and C respectively. If the Frenet (1, 3)-normal plane at each point
c(s) of C coincides with the Frenet (1, 3)-normal plane at each point c(s) of C coincides with
the Frenet (1, 3)-normal plane at corresponding point c(s)  c( (s)) of C for all s  L, then

C is called a (1, 3)-Bertrand curve in E4 and C is called a (1, 3)-Bertrand mate of C. We
obtain a characterization of (1, 3)-Bertrand curve, that is, we obtain Theorem B.
22.6 Keyword
Bertrand curve: If n  4, then no C -special Frenet curve in E is a Bertrand curve.
n

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