Manuscripta Mathematica

, Volume 136, Issue 1–2, pp 115–141 | Cite as

Cut loci and conjugate loci on Liouville surfaces

  • Jin-ichi Itoh
  • Kazuyoshi KiyoharaEmail author


In the earlier paper (Itoh and Kiyohara, Manuscr Math 114:247–264, 2004), we showed that the cut locus of a general point on two-dimensional ellipsoid is a segment of a curvature line and proved “Jacobi’s last geometric statement” on the singularities of the conjugate locus. In the present paper, we show that a wider class of Liouville surfaces possess such simple cut loci and conjugate loci. The results include the determination of cut loci and the set of poles on two-sheeted hyperboloids and elliptic paraboloids.

Mathematics Subject Classification (2000)

Primary 53C22 Secondary 53A05 


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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of EducationKumamoto UniversityKumamotoJapan
  2. 2.Department of Mathematics, Faculty of ScienceOkayama UniversityOkayamaJapan

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