Page 241 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Proof. Let a = t < t < · · · < t = b be a partition of [a, b] so that the diameter of e ([t , t ]) is less
0
1
1
i
i-1
n
than 2, i.e., e restricted to each subinterval maps into a semi-circle. Such a partition exists since
1
e is uniformly continuous on [a, b]. Choose (a) so that (3) holds at a, and proceed by induction
1
on i: if  is defined at t then there is a unique continuous extension so that (3) holds. If  is any
i
)

other continuous function satisfying (3), then k  (1/2 )(   is a continuous integer-valued
function, hence is constant. Finally, e = (–sin , cos ) hence
2



e  ke   ( sin ,cos ),

1
2
and we obtain  = k.
20.3.2 Global Theory
n
n
Definition 7. A curve :[a,b]   is closed if  (k) (a)   (k) (b). A closed curve :[a,b]   is
simple if  (a,b) is one-to-one. The rotation number of a smooth closed curve is:
1
n   (a)   (b)  , ...(4)

2
where  is the function defined in Proposition 2.
We note that the rotation number is always an integer. For reference, we also note that the
rotation number of a curve is the winding number of the map e . Finally, in view of the last
1
statement in Proposition 2, we have:
1
n   k ds.

2 [0,L]
2
Theorem 5 (Rotation Theorem). Let  :[0,L]  be a smooth, regular, simple, closed curve.
Then n   In particular,
1.

1 1.

2 [0,L] k ds  
For the proof, we will need the following technical lemma. We say that a set    is star-
n
shaped with respect to x  if for every y  the line segment x y lies in .
0
0
1
,
Lemma 1. Let    Rn be star-shaped with respect to x  and let e :    be a continuous
n
0
function. Then there exists a continuous function :    such that:
e = (cos , sin ). ...(5)
2
Moreover, if  is another continuous function satisfying (5), then     k where k is a
constant integer.
In fact, it is sufficient to assume that  is simply connected, but we will not prove this more
general result here.
Proof. Define (x ) so that (5) holds at x . For each x  , define  continuously along the line
0
0
segment x x as in the proof of Proposition 2. Since  is star-shaped with respect to x , this
0
0

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