18.950 Differential Geometry, Lecture 10 Notes | Massachusetts Institute of Technology - Edubirdie

University:
Massachusetts Institute of Technology
Course:
18.950 | Differential Geometry
Academic year:

2020
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499

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1
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do3rang8n

Lecture 10 Lemma 10.1 (Sturm-Hurwitz). Let f : R → R be a continuous 2π-periodic function such that � 2π � 2π � 2π f (t) dt = 0, f (t) cos(t) dt = 0, f (t) sin(t) dt = 0. 0 0 0 Then f has at least four zeros in the region [0, 2π). Lemma 10.2. Let h be a smooth 2π-periodic function. Then h(t) + h�� (t) has at least four critical points (points where its derivative vanishes) in the region [0, 2π). Lemma 10.3. Take a simple closed curve whose curvature is everywhere positive. By reparametrizing in a suitable way, one can achieve that the curve has period 2π and satisﬁes c� (t) = (cos(t), sin(t)). �c� (t)� In that case, 1 κ(t) = � . �c (t)� Theorem 10.4 (Four Vertex theorem, strictly convex version). Take a sim ple closed curve whose curvature is everywhere positive. Then there are at least four points where κ� (t) = 0.

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