The conjugate locus in convex 3-manifolds

Thomas Waters, Matthew Alan Cherrie

Research output: Contribution to journalArticlepeer-review

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In this paper we study the conjugate locus in convex manifolds. Our main tool is Jacobi fields, which we use to define a special coordinate system on the unit sphere of the tangent space; this provides a natural coordinate system to study and classify the singularities of the conjugate locus. We pay particular attention to 3-dimensional manifolds, and describe a novel method for determining conjugate points. We then make a study of a special case: the 3-dimensional (quadraxial) ellipsoid. We emphasise the similarities with the focal sets of 2-dimensional ellipsoids.
Original languageEnglish
Pages (from-to)17-30
Number of pages14
JournalNew Zealand Journal of Mathematics
Publication statusPublished - 1 Jul 2023


  • conjugate locus
  • ellipsoid
  • jacobi field
  • singularity


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