##plugins.themes.bootstrap3.article.sidebar##
##plugins.themes.bootstrap3.article.main##
Abstract
In this paper we construct a new class of surfaces whose geodesic flow is integrable (in the sense of Liouville). We do so by generalizing the notion of tubes about curves to 3-dimensional manifolds, and using Jacobi fields we derive conditions under which the metric of the generalized tubular sub-manifold admits an ignorable coordinate. Some examples are given, demonstrating that these special surfaces can be quite elaborate and varied.
##plugins.themes.bootstrap3.article.details##
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution (CC-BY) 4.0 License that allows others to share the work with an acknowledgment of the work’s authorship and initial publication in this journal.
Provided they are the owners of the copyright to their work, authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal’s published version of the work (e.g., post it to an institutional repository, in a journal or publish it in a book), with an acknowledgment of its initial publication in this journal.
Authors are permitted and encouraged to post their work online (e.g., in institutional repositories, disciplinary repositories, or on their website) prior to and during the submission process.
References
2. A. Bolsinov. Integrable geodesic flows on Riemannian manifolds. Journal of Mathematical Sciences, 123(4):4185–4197, 2004.
3. Alexey V. Bolsinov, Boˇzidar Jovanović. Integrable geodesic flows on Riemannian manifolds: construction and obstructions. In Contemporary geometry and related topics, pages 57–103. World Sci. Publ., River Edge, NJ, 2004.
4. Thierry Combot, Thomas Waters. Integrability conditions of geodesic flow on homogeneous Monge manifolds. Ergodic Theory Dynam. Systems, 35(1):111–127, 2015.
5. Manfredo Perdigão do Carmo. Riemannian geometry. Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty.
6. Holger R. Dullin, Vladimir S. Matveev. A new integrable system on the sphere. Math. Res. Lett., 11(5-6):715–722, 2004.
7. Alfred Gray. Tubes. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1990.
8. Eugene Gutkin. Curvatures, volumes and norms of derivatives for curves in Riemannian manifolds. J. Geom. Phys., 61(11):2147–2161, 2011.
9. Jorge V. José, Eugene J. Saletan. Classical dynamics. Cambridge University Press, Cambridge, 1998. A contemporary approach.
10. Wilhelm P. A. Klingenberg. Riemannian geometry, volume 1 of De Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, second edition, 1995.
11. Takashi Sakai. Riemannian geometry, volume 149 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1996. Translated from the 1992 Japanese original by the author.
12. Serge Tabachnikov. Geometry and billiards, volume 30 of Student Mathematical Library. American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005.
13. Michiko Tamura. Surfaces which contain helical geodesics in the 3-sphere. Mem. Fac. Sci. Eng. Shimane Univ. Ser. B Math. Sci., 37:59–65, 2004.
14. T. J. Waters. Non-integrability of geodesic flow on certain algebraic surfaces. Phys. Lett. A, 376(17):1442–1445, 2012.
15. Thomas Waters. Bifurcations of the conjugate locus. J. Geom. Phys., 119:1–8, 2017.
16. Thomas J. Waters. Regular and irregular geodesics on spherical harmonic surfaces. Phys. D, 241(5):543–552, 2012.