凸曲面上的卵形线的长

THE LENGTH OF AN OVALONAC ON ACONVEX SURFACE

若一凸曲面上一条简单封闭曲线围成一个同胚于平面的自凸域，则该曲线称为卵形线。Ａ．Ｖ，Ｐｏｇｏｒｅｌｏｖ曾证明，若正则凸闭曲面Ｆ的Ｇａｕｓｓ曲率不小于正的常量Ｋ，则Ｆ上的简单封闭测地线的长不大于作者将这定理推广成为如下的定理１若凸曲面Ｆ的比值曲率≥Ｋ，Ｋ一正的常量，则Ｆ上的卵形线的长不大于．

A simple closed curve on a Surface F is called an oval if and only if the region on F bounded bythis curve is self Convex and homeamorphic to a plane,The following theorem,which is a generaliza-tion of a theorem of Pogorelov,is prove.Theorem If F is a convex surface with ratio curvature≥K,then an oval on F must havelength at most