关于球面闭曲线的一个不等式

A inequality about the closed curve on a sphere

设γ为 S2 ( 1 )上的光滑闭曲线 ,k( s)为γ的曲率 ,L (γ)为γ的长度 ,A为γ分 S2 ( 1 )的两个区域的面积之一 ,文章得出∫γk2 ds≥ L(γ) + ( 2π -A) 2L(γ) ,且等号成立的条件是γ为 S2 ( 1 )

Let γ is a smoothed curve on a sphere S 2(1), k(s) is the curvature, L(γ) is the length , A is the area which is one part of the S 2(1) divided by γ , we prove ∫γk 2ds≥L(γ)+(2π-A) 2L(γ) and the equality is held if and only if γ is a circle on the S 2(1) .