Geometriae Dedicata

, Volume 169, Issue 1, pp 83–98 | Cite as

Spherical geometry and integrable systems

Original paper


We prove that the cosine law for spherical triangles and spherical tetrahedra defines integrable systems, both in the sense of multidimensional consistency and in the sense of dynamical systems.


Spherical triangle Spherical simplex Cosine law  Sine law  Integrable systems Multidimensional consistency Discrete Darboux system  Euler top Hirota–Kimura discretization 

Mathematics Subject Classification (2000)

51M10 37J35 37K25 37K10 



The authors are partly supported by DFG (Deutsche Forschungsgemeinschaft) in the frame of Sonderforschungsbereich/Transregio 109 “Discretization in Geometry and Dynamics”.


  1. 1.
    Bobenko, A.I., Suris, Yu.B.: Discrete Differential Geometry. Integrable structure, Graduate Studies in Mathematics, vol. 98. AMS, Providence (2008)Google Scholar
  2. 2.
    Bogdanov, L.V., Konopelchenko, B.G.: Lattice and \(q\)-difference Darboux-Zakharov-Manakov systems via \(\bar{\partial }\)-dressing method. J. Phys. A 28:L173–Ll78 (1995)Google Scholar
  3. 3.
    Derevnin, D.A., Mednykh, A.D., Pashkevich, M.G.: On the volume of symmetric tetrahedron. Sib. Math. Jour. 45(5):840–848 (2004)Google Scholar
  4. 4.
    Doliwa, A.: The C-(symmetric) quadrilateral lattice, its transformations and the algebro-geometric construction. J. Geom. Phys. 60, 690–707 (2010)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Hirota, R., Kimura, K.: Discretization of the Euler top. J. Phys. Soc. Jpn. 69, 627–630 (2000)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Hone, A.N.W., Petrera, M.: Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras. J. Geom. Mech 1(1), 55–85 (2009)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Jonas, H.: Deutung einer birationalen Raumtransformation im Bereiche der sphärischen Trigonometrie. Math. Nachr 6, 303–314 (1951)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Kashaev, R.M.: On discrete three-dimensional equations associated with the local Yang-Baxter relation. Lett. Math. Phys. 35, 389–397 (1996)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Konopelchenko, B.G., Schief, W.K.: Three-dimensional integrable lattices in Euclidean spaces: conjugacy and orthogonality. Proc. R. Soc. Lond. A 454, 3075–3104 (1998)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Kolpakov, A.A., Mednykh, A.D., Pashkevich, M.G.: Volume Formula For a \(\mathbb{Z}_2\)-Symmetric Spherical Tetrahedron Through its Edge Lengths. Arkiv för Matematik, 2013, published online at doi: 10.1007/s11512-011-0148-2 (to appear)
  11. 11.
    Luo, F.: On a problem of Fenchel. Geom. Dedicata 64, 277–282 (1997)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Luo, F.: Volume and angle structures on 3-manifolds. Asian J. Math. 11(4), 555–566 (2007)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Luo, F.: 3-Dimensional Schläfli formula and its generalization. Commun. Contemp. Math. 10, 835–842 (2008)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Petrera, M., Pfadler, A., Suris, YuB: On integrability of Hirota-Kimura type discretizations. Experimental study of the discrete Clebsch system. Exp. Math. 18(2), 223–247 (2009)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Petrera, M., Pfadler, A., Suris, YuB: On integrability of Hirota-Kimura type discretizations. Reg. Chaot. Dyn. 16(3–4), 245–289 (2011)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Petrera, M., Suris, YuB: On the Hamiltonian structure of Hirota-Kimura discretization of the Euler top. Math. Nachr. 283(11), 1654–1663 (2011)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Reyman, A.G., Semenov-Tian-Shansky, M.A.: Group theoretical methods in the theory of finite-dimensional integrable systems. In: Dynamical Systems VII. Springer, Berlin (1994)Google Scholar
  18. 18.
    Schief, W.K.: Lattice geometry of the giscrete Darboux, KP, BKP and CKP equations. Menelaus and Carnot theorems. J. Nonlinear Math. Phys. 10(2), 194–208 (2003)MathSciNetGoogle Scholar
  19. 19.
    Sergeev, S.M.: Solutions of the functional tetrahedron equation connected with the local Yang-Baxter equation for the ferro-electric condition. Lett. Math. Phys. 45, 113–119 (1998)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Vinberg, E.B. (Ed.): Geometry II, Encyclopaedia of Mathematical Sciences, vol. 29. Springer, Berlin (1993)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Institut für Mathematik, MA 7-2Technische Universität BerlinBerlinGermany

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