Bifurcations of the conjugate locus
arXiv:1704.02001 · doi:10.1016/j.geomphys.2017.04.003
Abstract
The conjugate locus of a point in a surface will have a certain number of cusps. As the point 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.
Accepted in Journal of Geometry and Physics April 2017