Journal of Dynamical and Control Systems

, Volume 15, Issue 1, pp 99–131 | Cite as

On Zermelo-Like Problems: Gauss–Bonnet Inequality and E. Hopf Theorem

  • Ulysse Serres


The aim of this paper is to describe the Zermelo navigation problem on Riemannian manifolds as a time-optimal control problem and give an efficient method of evaluating its control curvature. We will show that, up to the change of the Riemannian metric on the manifold, the control curvature of the Zermelo problem has a simple to handle expression which naturally leads to a generalization of the classical Gauss–Bonnet formula in the form of an inequality. This Gauss–Bonnet inequality allows one to generalize the Zermelo problems and obtain a theorem of E. Hopf that establishes the flatness of Riemannian tori without conjugate points.

Key words and phrases

Conjugate points control curvature feedback transformation Gauss–Bonnet formula Riemannian manifold Zermelo navigation problem 

2000 Mathematics Subject Classification

34K35 37C10 37E35 53B40 53C22 93C15 


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  1. 1.
    A. A. Agrachev, Feedback-invariant optimal control theory and differential geometry, II. Jacobi curves for singular extremals. J. Dynam. Control Systems 4 (1998), No. 4, 583–604.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A. A. Agrachev and N. N. Chtcherbakova, Hamiltonian systems of negative curvature are hyperbolic. Russ. Math. Dokl. 400 (2005), 295–298.Google Scholar
  3. 3.
    A. A. Agrachev, N. N. Chtcherbakova, and I. Zelenko, On curvatures and focal points of dynamical lagrangian distributions and their reductions by first integrals. J. Dynam. Control Systems 11 (2005), No. 3, 297–327.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    A. A. Agrachev and R. V. Gamkrelidze, Feedback-invariant optimal control theory and differential geometry, I. Regular extremals. J. Dynam. Control Systems 3 (1997), No. 3, 343–389.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    A. A. Agrachev and Yu. L. Sachkov, Control theory from the geometric viewpoint. Encycl. Math. Sci. 87, Springer-Verlag (2004).Google Scholar
  6. 6.
    A. A. Agrachev and I. Zelenko, Geometry of Jacobi curves, I. J. Dynam. Control Systems 8 (2002), No. 1, 93–140.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    D. V. Anosov and Ya. G. Sinai, Some smooth ergodic systems. Russ. Math. Surv. 22 (1967), No. 5, 103–167.CrossRefMathSciNetGoogle Scholar
  8. 8.
    V. I. Arnold, Some remarks on flows of line elements and frames. Sov. Math. Dokl. 2 (1961), 562–564.Google Scholar
  9. 9.
    W. Ballmann, M. Brin, and K. Burns, On surfaces with no conjugate points. J. Differ. Geom. 25 (1987), No. 2, 249–273.zbMATHMathSciNetGoogle Scholar
  10. 10.
    D. Bao, C. Robles, and Z. Shen, Zermelo navigation problem on Riemannian manifolds. J. Differ. Geom. 66 (2004), 391–449.MathSciNetGoogle Scholar
  11. 11.
    D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat. Geom. Funct. Anal. 4 (1994), No. 3, 259–269.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    C. Carathéodory, Calculus of variations. Chelsea, New York (1989).Google Scholar
  13. 13.
    S.-S. Chern and Z. Shen, Riemann–Finsler geometry. Nankai Tracts Math. 6, World Scientific (2005).Google Scholar
  14. 14.
    L. W. Green, Surfaces without conjugate points. Trans. Amer. Math. Soc. 76 (1954), No. 3, 529–546.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    M. Gromov, Groups of polynomial growth and expanding maps (with an appendix by J. Tits). Publ. Math. IHES 53 (1981), 53–78.zbMATHMathSciNetGoogle Scholar
  16. 16.
    B. Hasselblatt and A. Katok, Introduction to the modern theory of dynamical systems. Cambridge Univ. Press, New York (1995).zbMATHGoogle Scholar
  17. 17.
    E. Hopf, Closed surfaces without conjugate points. Proc. Natl. Acad. Sci. USA 34 (1948), No. 2, 47–51.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    P. Przytycki, Hamiltonowskie podejście do niezmienników i krzywizny [in Polish]. Master thesis, Warsaw University (2004).Google Scholar
  19. 19.
    U. Serres, Géométrie et classification par feedback des systèmes de contrôle non linéaires de basse dimension. Ph.D. thesis, Université de Bourgogne, Dijon (2006);
  20. 20.
    U. Serres, On the curvature of two-dimensional optimal control systems and Zermelo’s navigation problem. J. Math. Sci. 135 (2006), No. 4, 3224–3243.CrossRefMathSciNetGoogle Scholar
  21. 21.
    S. Sternberg, Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang–Mills field. Proc. Natl. Acad. Sci. USA 74 (1977), No. 12, 5253–5254.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    E. Zermelo, Über das Navigationsproblem bei ruhender oder veränderlicher windverteilung. Z. Angew. Math. Mech. 11 (1931), No. 2, 114–124.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Institut Élie Cartan de Nancy UMR 7502Nancy-Université, CNRS, INRIAVandœuvre-lès-Nancy CedexFrance

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