Page 219 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes N(s) = ( x (s),x (s)). ' 2 ' 1 ...(3)

Expressing the last two equations of (1) in components, we have

"
x   kx ' 2 ...(4)
1
'
"
x  kx . ...(5)
1
2
These equations are equivalent to (1).
Since T is a unit vector, we can put
T(s) = (cos (s), sin (s)), ...(6)
so that t(s) is the angle of inclination of T with the x -axis. Then
1
N(s) = (–sin (s), cos (s)), ...(7)

and (1) gives

d  k(s) ...(8)
ds
This gives a geometrical interpretation of k(s).
A curve C is called simple if it does not intersect itself. One of the most important theorems in
global differential geometry is the theorem on turning tangents:
Theorem 3. For a simple closed plane curve, we have

1 1.
2   k ds  
To prove this theorem we give a geometrical interpretation of the integral at the left-hand side
of (3). By (8)

1 1 d .
2   k ds  2   
But , as the angle of inclination of (s), is only defined up to an integral multiple of 2, and this
integral has to be studied with care.
Let O be a fixed point in the plane. Denote by  the unit circle about O; it is oriented by the
orientation of the plane. The tangential mapping or Gauss mapping

g : C   ...(9)
is defined by sending the point x(s) of C to the point T(s) of . In other words, g(P), P  C, is the
end-point of the unit vector through O parallel to the unit tangent vector to C at P. Clearly, g is
a continuous mapping. If C is closed, it is intuitively clear that when a point goes along C once
its image point under g goes along Û a number of times. This integer is called the rotation index
of C. It is to be defined rigorously as follows:
We consider O to be the origin of our coordinate system. As above we denote by (s) the angle
of inclination of T(s) with the x -axis. In order to make the angle uniquely determined, we
1
suppose O  (s) < 2. But (s) is not necessarily continuous. For in every neighborhood of s at
0
which (s ) = 0, there may be values of (s) differing from 2 by arbitrarily small quantities.
c

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