Page 221 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes where k(p) is an integer. Let q0 be any point on the segment mp . Draw the segment qq 0
0
parallel to pp , with q on mp. The function (q) t(q )    0 is continuous in q along mp and equals
0
O when q coincides with m. Since d(q, q ) < , it follows from (11) that (q) t(q )    0  .  In
0
.
particular, for q = p this gives (p)     (p )   Combining this with (12), we get k(p) = 0. Thus
0
0
0
 
we have proved that (p) is continuous in 4, as asserted above. Since (p)   (p) mod 2, it is
 
clear that (p) is differentiable.
 
Now let A(O,O), B(O,L), D(L,L) be the vertices of . The rotation index  of C is, by (10), defined
by the line integral
d 
2    AD .
Since ˜t(p) is defined in 4, we have



  AD d    AB d    BD d .  
To evaluate the line integrals at the right-hand side, we suppose the origin O to be the point x(O)
and C to lie in the upper half-plane and to be tangent to the x -axis at O. This is always possible
1
for we only have to take x(O) to be the point on C at which the x -coordinate is a minimum. Then
2
the x -axis is either in the direction of the tangent vector to C at O or opposite to it. We can
1
assume the former case, by reversing the orientation of C if necessary. The line integral along
AB is then equal to the angle rotated by OP as P goes once along C. Since C lies in the upper
half-plane, the vector OP never points downward. It follows that the integral along AB is equal

to . On the other hand, the line integral along BD is the angle rotated by PO as P goes once
along C. Since the vector PO never points upward, this integral is also equal to . Hence, their
sum is 2 and the rotation index  is +1. Since we may have reversed the orientation of C, the
rotation index is ±1 in general.

Exercise 21: Consider the plane curve x(t) = (t, f(t)). Use the Frenet formulas in (1) to prove that
its curvature is given by

 f
k(t) = . ...(13)
(1 f )

2 3/2

Exercise 22: Draw closed plane curves with rotation indices 0, –2, +3 respectively.
Exercise 23: The theorem on turning tangents is also valid when the simple closed curve C has
“corners.” Give the theorem when C is a triangle consisting of three arcs. Observe that the
theorem contains as a special case the theorem on the sum of angles of a rectilinear triangle.
Exercise 24: Give in detail the proof of the existence of  = (p ) used in the proof of the theorem
0
on turning tangents.  =  (p ) .
0

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