# Bifurcations of the conjugate locus

## TL;DR

This paper investigates the bifurcations of the conjugate locus on surfaces, explaining how cusps appear or disappear as a point moves, through analysis of the exponential map's higher derivatives and local geometric structures.

## Contribution

It introduces a new framework linking cusp bifurcations to higher derivatives of the exponential map and classifies these bifurcations based on local conjugate locus structures.

## Key findings

- Derived equations for higher derivatives of the exponential map.
- Classified bifurcations of cusps in conjugate loci.
- Provided an intuitive geometric interpretation of bifurcations.

## Abstract

The conjugate locus of a point $p$ in a surface $\mathcal{S}$ will have a certain number of cusps. As the point $p$ is moved in the surface the conjugate locus may spontaneously gain or lose cusps. In this paper we explain this `bifurcation' in terms of the vanishing of higher derivatives of the exponential map; we derive simple equations for these higher derivatives in terms of scalar invariants; we classify the bifurcations of cusps in terms of the local structure of the conjugate locus; and we describe an intuitive picture of the bifurcation as the intersection between certain contours in the tangent plane.