Title: The conjugate locus on convex surfaces Show Chinese title

Title: 共轭点集在凸曲面上

Abstract: The conjugate locus of a point on a surface is the envelope of geodesics emanating radially from that point. In this paper we show that the conjugate loci of generic points on convex surfaces satisfy a simple relationship between the rotation index and the number of cusps. As a consequence we prove the `vierspitzensatz': the conjugate locus of a generic point on a convex surface must have at least four cusps. Along the way we prove certain results about evolutes in the plane and geodesic curvature. (Note: this is a corrected version of the original paper, see comment on page 5 and Appendix B). Show Chinese abstract

Abstract: 点在曲面上的共轭点丛是沿该点径向发出的测地线的包络面。 本文中，我们证明了凸面上一般点的共轭点丛在旋转指标和尖点数量之间满足一个简单的关系。 作为结果，我们证明了“四尖点定理”：凸面上一般点的共轭点丛必须至少有四个尖点。 在这一过程中，我们证明了平面上的渐近线和测地曲率的一些结果。 （注：这是原文的修正版，参见第5页注释及附录B）。