关于空间曲缐多边形的全曲率

<正> 在另一文内作者证明了这样定理:设一关闭挠曲缐 C 有一角点它的内角是θ,则它的全曲率∮_(c)kds≥π+θ.这结果可以看做关于关闭挠曲线全曲率的 Fenchel 定理的推广.从这结果很自然会引起一个问题,就是如果所论闭曲缐的角点多于一个,则

ON THE INTEGRAL CURVATURE OF A CURVILINEAR POLYGON

We say,for simplicity,a closed space curve with finite angular pointsa curvilinear polygon.The purpose of this paper is to establish the fol-lowingTheorem:For the integral of the frst curvature of any curvilinearpolygon C in m-dimensional Euclidean space S_m holds the followinginequality:(?)where θ_1,θ_2,…,θ_n are the interior angles of C.The equality holds onlyfor a plane convex curvilinear polygon.Corollary 1.For the integral curvature of any curvilinear polygon C inordinary space S_3 holds the following inequality: (?)Corollary 2.The integral of the first curvature of any closed spacecurve in space S_m is not less than 2π.The corresponding theorem of corollary 2 for a closed curve in ordinaryspace S_3 is due to W.Fenchel.Corollary 3.The integral of the first curvature of any closed curvewith an angular point in space S_m is not less than π.Corollary 4.The integral of the first curvature of any closed curvewith one k-multiple point in space S_m is not less than kπ.