Lecture 9 Theorem 9.1 (Hopf Umlaufsatz; sketch proof). Let c be a simple closed curve. Then κtot (c) = ±2π. The sign here can be determined as follows. Let t be a point where c2 (t) reaches its (global) minimum. Then the sign of κtot (c) equals that of c�1 (t). Definition 9.2. Let c be a simple closed curve. We say that c is convex if the following holds. Whenever c is tangent to some line {a1 x1 +a2 x2 = b} in the plane, it is entirely contained in one of the two half-planes {a1 x1 +a2 x2 ≤ b}, {a1 x1 + a2 x2 ≥ b}. Proposition 9.3 (partial proof). A simple closed curve is convex if and only if its curvature never changes sign. Corollary 9.4 (sketch proof). Let c be a closed curve of period T . Then � T |κ(t)| �c� (t)� dt ≥ 2π. 0 Here is a useful generalization of the Umlaufsatz. Take a closed curve c of period T . Suppose that c takes on the same value at most twice in [0, T ). Moreover, for any 0 ≤ s < t < T such that c(s) = c(t), we also require c� (s) and c� (t) to be linearly independent. In that case, we say that c has normal self-intersections. Theorem 9.5 (Whitney; no proof). Let c be a closed curve with normal self-intersections. Assume that it is parametrized in such a way that c2 (t) reaches a global minimum at t = 0. Then � κtot (c)/2π = sign c�1 (0) − sign det(c� (s), c� (t)), (s,t) where the sum is over all 0 ≤ s < t < T with c(s) = c(t).