論文

査読有り
2006年

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

MATHEMATICS OF COMPUTATION
  • Robert Sinclair
  • ,
  • Minoru Tanaka

75
256
開始ページ
1779
終了ページ
1808
記述言語
英語
掲載種別
研究論文(学術雑誌)
DOI
10.1090/S0025-5718-06-01924-7
出版者・発行元
AMER MATHEMATICAL SOC

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) = const or x(2) = 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.

Web of Science ® 被引用回数 : 5

リンク情報
DOI
https://doi.org/10.1090/S0025-5718-06-01924-7
Web of Science
https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcAuth=JSTA_CEL&SrcApp=J_Gate_JST&DestLinkType=FullRecord&KeyUT=WOS:000241378700010&DestApp=WOS_CPL
URL
http://orcid.org/0000-0002-9646-4322
ID情報
  • DOI : 10.1090/S0025-5718-06-01924-7
  • ISSN : 0025-5718
  • ORCIDのPut Code : 15293301
  • Web of Science ID : WOS:000241378700010

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