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Jacobi’s last geometric statement extends to a wider class of Liouville surfaces. (English) Zbl 1124.53015

The statement that the conjugate locus from a point has exactly four cusps and the corresponding cut locus consists of only one topological segment with two endpoints is known as Jacobi’s last geometric statement [cf. the authors, The Cut Locus of a 2-Sphere of Revolution and Toponogov’s Comparison Theorem, (to appear)]. The authors present a conjecture – in the form of a sufficient rather than a necessary condition – that this statement can be extended from the original case of an ellipsoid to a “wider class of compact real analytic Liouville surfaces diffeomorphic to the two-sphere \(S^2\) if the Gaussian curvature is everywhere positive and has exactly six critical points, these being two saddles, two global minima and two global maxima”. A variety of (standard) computational techniques are employed as numerical evidence of it, which also provides some useful insights into the construction of the pure mathematical proofs. Also the confirmation of a relevant result a forthcoming paper by J. I. Itoh and K. Kiyohara is announced.

MSC:

53C20 Global Riemannian geometry, including pinching
53A05 Surfaces in Euclidean and related spaces

Software:

Thaw; Loki
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Full Text: DOI

References:

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