Lecture 8 Definition 8.1. A closed curve of period T is a regular curve c : R → R2 such that c(t + T ) = c(t) for all t. We say that c is simple if it has no selﬁntersections. This means that for all 0 ≤ s < t < T , we have c(s) = � c(t). Theorem 8.2 (Jordan curve theorem; very sketchy proof). Let c be a simple closed curve. Then, the complement of the image of c is the disjoint union of two connected open subsets, one bounded (the inside) and one unbounded (the outside) The hard step in the proof is to show that the inside and outside are not connected to each other. For that, one uses winding numbers. Points in the inside have winding number �= 0, and points in the outside have winding number 0. On the other hand, the winding number is locally constant. Definition 8.3. The total curvature of a closed curve is deﬁned to be � T κtot (c) = κ(t) �c� (t)� dt. 0 Physically, κtot is a dimensionless quantity. Lemma 8.4 (partial proof). Let c be a closed curve of period T , and set �T L = 0 �c� (t)� dt. Let d be the unit speed reparametrization of c. Then d is again a closed curve, of period L. Moreover, the total curvature of d is the same as that of c. Proposition 8.5. κtot (c)/2π is the degree of f (t) = c� (t)/�c� (t)�. In par ticular, it is always an integer. We call it the rotation number of the curve (not to be confused with the winding number: the rotation number is the winding number of c� (t) around 0). Corollary 8.6. Let c be a closed curve of period T . Suppose that there are only ﬁnitely many points 0 ≤ t1 < t2 < · · · < tm < T where c�2 (tk ) = 0, c�1 (tk ) > 0, and that any such point satisﬁes κ(tk ) �= 0. Then, the rotation number is m � tot κ (c)/2π = sign(κ(tk )). k=1