The conjugate locus on convex surfaces
Differential Geometry
2025-04-10 v2
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).
Cite
@article{arxiv.1806.00278,
title = {The conjugate locus on convex surfaces},
author = {Thomas Waters},
journal= {arXiv preprint arXiv:1806.00278},
year = {2025}
}
Comments
Accepted Geometriae Dedicata May 2018. New version posted March 2025 correcting Section 2.2 (details in added Appendix B), main results unaffected