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arXiv · 1704.02001

Bifurcations of the conjugate locus

Thomas Waters
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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.

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Thomas Waters. 2017-04-06. Bifurcations of the conjugate locus. https://doi.org/10.1016/j.geomphys.2017.04.003

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