Sinclair, R. On the last geometric statement of Jacobi. (English) Zbl 1073.53007 Exp. Math. 12, No. 4, 477-485 (2003). The last geometric statement of Jacobi concerns the structure of the conjugate locus of a point in an ellipsoid. The claim is that this always has four cusps. A more general version would include the statement that the cut locus of a point is an arc on the curvature line through the antipodal point. The present paper contains numerical experiments which support these statements. A number of nice figures are given which indicate the four cusps. For a description of a computer tool called Thaw see J. Itoh and the author in [Exp. Math. 13, 309–325 (2004)]. Both statements were later proven without a computer by J. Itoh and K. Kiyohara in [Manuscr. Math. 114, No. 2, 247–264 (2004; Zbl 1076.53042)]. Reviewer: Wolfgang Kühnel (Stuttgart) Cited in 1 ReviewCited in 9 Documents MSC: 53A05 Surfaces in Euclidean and related spaces 53-04 Software, source code, etc. for problems pertaining to differential geometry Keywords:caustic; cut locus; conjugate locus; triaxial ellipsoid; cusps Citations:Zbl 1076.53042 Software:Thaw; Loki PDFBibTeX XMLCite \textit{R. Sinclair}, Exp. Math. 12, No. 4, 477--485 (2003; Zbl 1073.53007) Full Text: DOI Euclid EuDML References: [1] Arnold V. I., Topological Invariants of Plane Curves and Caustics 5 (1994) [2] Arnold V. I., The Arnoldfest 24 pp 39– (1999) [3] Benamou J. -D., Journal of Computational Physics 162 (1) pp 132– (2000) · Zbl 0958.65076 [4] Berger M., University Lecture Series 17, in: Riemannian Geometry During the Second Half of the Twentieth Century (2000) [5] Bolsinov A. V., Integrable Geodesic Flows on Two Dimensional Surfaces (1999) [6] von Braunmüuhl A., Mathematische Annalen pp 557– (1879) · JFM 11.0539.01 [7] von Braunmuühl A., Mathematische Annalen pp 557– (1882) · JFM 14.0689.03 [8] do Carmo M. P., Riemannian Geometry. (1992) [9] Duistermaat J. J., Communications on Pure and Applied Mathematics pp 207– (1974) · Zbl 0285.35010 [10] Ehlers J., Journal of Mathematical Physics 41 (6) pp 3344– (2000) · Zbl 0974.58037 [11] Engquist B., Journal of Computational Physics 178 (2) pp 373– (2002) · Zbl 0996.78001 [12] Featherstone W. E., Geomatics Research Australasia 64 pp 65– (1996) [13] Fomel S., Proceedings of the National Academy of Sciences of the USA 99 (11) pp 7329– (2002) · Zbl 1002.65113 [14] Giaquinta M., Calculus of Variations I, Grundlehren der mathematischen Wissenschaften 310 (1996) [15] Gjoystdal H., Studia Geophysica et Geodaetica 46 (2) pp 113– (2002) [16] Hanyga A., Symplectic Singularities and Geometry of Gauge Fields 39 pp 57– (1997) · Zbl 0903.41018 [17] Itoh J. -I., ”Thaw: A Tool for Approximating Cut Loci on a Triangulation of a Surface.” (2002) [18] Joets A., Experimental Mathematics 8 (1) pp 49– (1999) [19] Jacobi C. G. J., C.G.J. Jacobi’s Gesammelte Werke, pp 46– (1884) [20] Jacobi C. G. J., C. G. J. Jacobi’s Gesammelte Werke 7 pp 72– (1891) · JFM 22.0022.02 [21] Klingenberg W., Riemannian Geometry. (1982) [22] Knoörrer H., In ventiones Mathematicae 59 pp 119– (1980) · Zbl 0431.53003 [23] Kobayashi S., Studies in Global Geometry and Analysis 4 pp 96– (1967) [24] Lambare G., Geophysical Journal International 125 (2) pp 584– (1996) [25] Longuet-Higgins M., Proceedings of the Royal Society of London, Series A 428 pp 283– (1990) · Zbl 0711.53005 [26] von Mangoldt H., Journal füur die reine und angewandte Mathematik (1) pp 23– (1881) [27] Margerin C. M., Differential Geometry: Riemannian Geometry 54 pp 465– (1991) [28] Poincarëe H., Transactions of the American Mathematical Society 6 pp 237– (1905) [29] Postnikov M. M., Geometry VI: Riemannian Geometry 91 (2001) [30] Sakai T., Riemannian Geometry 149 (1996) · Zbl 0886.53002 [31] Sinclair R., Experimental Math-ematics 11 (1) pp 1– (2002) · Zbl 1052.53001 [32] Struik D. J., Lectures on Classical Differential Geometry, (1961) · Zbl 0105.14707 [33] Wolter F. -E., Archiv der Mathematik 32 pp 92– (1979) · Zbl 0409.53032 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.