The conjugate locus on convex surfaces

TL;DR

This paper investigates the properties of conjugate loci on convex surfaces, establishing a relationship between geometric features and proving that such loci must have at least four cusps for generic points.

Contribution

It introduces a simple relationship between rotation index and cusps of conjugate loci on convex surfaces and proves the minimal cusp count, extending understanding of geodesic behavior.

Findings

Conjugate loci satisfy a specific relationship between rotation index and cusps.

The conjugate locus of a generic point on a convex surface has at least four cusps.

Results about evolutes and geodesic curvature are established.

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).