Page 233 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Proof. Let X(t) be the matrix whose columns are the vectors e (t), . . ., e (t) of the Frenet frame
n
1
along f. Consider the system of ODE’s,
X’(t) = –X(t)(t),
with initial conditions X(0) = I, where (t) is the skew symmetric matrix of curvatures. By a
standard result in ODE’s, there is a unique solution X(t).
We claim that X(t) is an orthogonal matrix. For this, note
that
(XX )’ = X’X + X(X )’ = –XX – X X T
T
T
T
T
T
= –XX + XX = 0.
T
T
Since X(0) = I, we get XX = I. If F(t) is the first column of X(t), we define the curve f by
T
s
f(s) = F(t)dt,

0
with s  ]a, b[. It is easily checked that f is a curve parametrized by arc length, with Frenet frame
X(s), and with curvatures  ’s.
i
19.2 Summary
Lemma 1. Let f: ]a, b[  E (or f : [a, b]  E ) be a curve of class C , with p  n. A family (e (t),
p
n
n
 1
. . ., e (t)) of orthonormal frames, where each e : ]a, b[  E is C continuous for i = 1, . . .,
n–i
n
i
n
n – 1 and en is C -continuous, is called a moving frame along f. Furthermore, a moving
1
frame (e (t), . . ., e (t)) along f so that for every k, with 1  k  n, the kth derivative f (t) of
(k)
n
1
f(t) is a linear combination of (e (t), . . ., e (t)) for every t  ]a, b[, is called a Frenet n-frame
1
k
or Frenet frame.
Lemma 2. Let f: ]a, b[  E (or f: [a, b]  E ) be a curve of class C , with p  n, so that the
n
n
p

derivatives f (t), . . ., f (n–1) (t) of f(t) are linearly independent for all t  ]a, b[. Then, there is
(1)
a unique Frenet n-frame (e (t), . . ., e (t)) satisfying the following conditions:
n
1
The k-frames (f (t), . . ., f (t)) and (e (t), . . ., e (t)) have the same orientation for all k,
(k)
(1)
 1 k
with 1  k  n – 1.
The frame (e (t), . . ., e (t)) has positive orientation.
 1 n
Let f: ]a, b[  E (or f: [a, b]  E ) be curve of class C , with p  n, so that the derivatives
n
n
p

f (t), . . ., f (n–1) (t) of f(t) are linearly independent for all t  ]a, b[ . Let h : E  E be a rigid
n
n
(1)
motion, and assume that the corresponding linear isometry is R. Let f = h o f. The following

properties hold:
For any moving frame (e (t), . . ., e (t)), the n-tuple (e (t), . . . , e (t)), where


 1 n 1 n
e (t) = R(e (t)), is a moving frame along ef, and we have

i
i
  ij (t) =  ij (t)) and  f'(t)  f'(t) .
For any orientation-preserving diffeomorphism  : ]c, d[  ]a, b[ (i.e., ’(t) > 0 for all


t  ]c, d[ ), if we write f  f r, then for any moving frame (e (t), . . ., e (t)) on f, the

n
1
n-tuple (e (t), . . . , e (t)), where e (t) = e ((t)), is a moving frame on f.




i
n
1
i
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