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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Jacobi's last geometric statement extends to a wider class of Liouville surfaces


Authors: Robert Sinclair and Minoru Tanaka
Journal: Math. Comp. 75 (2006), 1779-1808
MSC (2000): Primary 53C20, 53--04; Secondary 53C25
Published electronically: June 14, 2006
MathSciNet review: 2240635
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Abstract | References | Similar Articles | Additional Information

Abstract: Numerical evidence is presented which strongly suggests that ``Jacobi's last geometric statement"--that the conjugate locus from a point has exactly four cusps and the corresponding cut locus consists of only one topological segment--holds for compact real analytic Liouville surfaces diffeomorphic to $ S^2$ if the Gaussian curvature is everywhere positive and has exactly six critical points, these being two saddles, two global minima, and two global maxima (as is the case for an ellipsoid). Our experiments suggest that this is a sufficient rather than a necessary condition. Furthermore, for compact real analytic Liouville surfaces diffeomorphic to $ S^2$ upon which the Gaussian curvature can be negative but has exactly six critical points, these being two saddles, two global minima, and two global maxima, it appears that the cut locus is always a subarc of a line given by $ x_1={\mathrm{const}}$ or $ x_2={\mathrm{const}}$, where $ (x_1,x_2)$ are canonical coordinates with respect to which the metric has the form $ (f_1(x_1)+f_2(x_2))(dx_1^2+dx_2^2)$. In the case of an ellipsoid, these curves are lines of curvature.


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Additional Information

Robert Sinclair
Affiliation: Department of Mathematical Sciences, Faculty of Science, University of the Ryukyus, Nishihara-Cho, Okinawa, 903–0213 Japan
Email: sinclair@math.u-ryukyu.ac.jp

Minoru Tanaka
Affiliation: Department of Mathematics, Tokai University, Hiratsuka City, Kanagawa, 259–1292 Japan
Email: m-tanaka@sm.u-tokai.ac.jp

DOI: http://dx.doi.org/10.1090/S0025-5718-06-01924-7
PII: S 0025-5718(06)01924-7
Keywords: Cut locus, conjugate locus, Liouville surface
Received by editor(s): October 4, 2004
Published electronically: June 14, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.