Page 247 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes
0   I  k   k ds K   2  0,

and it follows that k  k  0, which by Theorem 7 implies that  is convex.
20.5 Summary


Definition 1. A parametrized curve is a smooth (C ) function : Ig ®  n . A curve is regular if

' g ¹ 0.
When the interval I is closed, we say that  is C on I if there is an interval J and a C 

function  on J which agrees with  on I.
n
n
 Definition 2. Let g : I ®  be a parametrized curve, and let b : J ®  be another
parametrized curve. We say that  is a reparametrization (orientation preserving
t
reparametrization) of  if there is a smooth map : J ® I with ' 0t > such that b = g t .

Definition 4. Let  be a regular curve. We say that  is parametrized by arclength if '  1

Note that this is equivalent to the condition that for all t  I = [a, b] we have:
t
L  [a,t]   a.

Furthermore, any regular curve can be parametrized by arclength. Indeed, if  is a regular
curve, then the function

t
s(t)    ' ,
a
3
'
Definition 5. Let  : I   be a curve in  . The unit vector T   is called the unit
3

-1
tangent of . The curvature  is the scalar    " . The unit vector N = k T’ is called the
principal normal. The binormal is the unit vector B = T × N. The positively oriented
orthonormal frame (T, N, B) is called the Frenet frame of .
Theorem 3. Let  be a regular curve with k  0. Then  is a straight line.

Proof. Since T'  k  0, it follows that T is constant and  is linear.
Theorem 4. Let  be a regular curve with k > 0, and  = 0. Then  is planar.


Proof. Since B’ = 0, B is constant. Thus the function      (0) . B vanishes identically:
 (0) 0,   T B  0.


It follows that remains in the plane through (0) perpendicular to B.
Theorem 5. Let be a regular curve with k constant and  = 0. Then  is a circle.

Proof. Let b =  + k N. Then
-1
1
'   T  ( kT   B)  0.

k
1
Thus,  is constant, and     k . It follows that  lies in the intersection between a

plane and a sphere, thus  is a circle.

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