R^3 上的 Fary-Milnor 定理

Abstract

Fary-Milnor 定理: 一條空間中的簡單封閉扭結(knotted simple closed curve)其總曲率必大於4π。也就是說，若γ:[0,l]→R^3為一和圓等倫(isotopic)且以弧長s為參數的曲線，k(s)為曲率，則∫|k(s)|ds>4π。 我們將說明這個定理適用於簡單封閉的多邊形。然後由於一條簡單封閉曲線總是能夠等倫於此曲線的某個內接多邊形，而且一個內接多邊形的總曲率不會超過原來曲線的總曲率，因此便證得了 Fary-Milnor 定理。

The Fary-Milnor theorem states that the total curvature of a knotted simple closed curve in R^3 is greater than 4π. That is, let γ:[0,l]→R^3 be isotopic to S^1 and be parametrized by arc length s with curvature k(s), then ∫|k(s)|ds>4π. We are going to show this theorem for simple closed ploygons since a simple closed curve of finite total curvature is isotopic to an inscribed polygon, and the total curvature of an inscribed polygon never exceeds that of the original curve.