ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

The Jacobi Equation for Horizontal Geodesics on a Nonholonomic Distribution and the Schouten Curvature Tensor


Victor Revoltovich Krym

PhD in Physics and Mathematics
St.-Petersburg Mathematical Society
Fontanka, 27, 191023, St.-Petersburg


The paper shows that if the distribution is defined on a manifold with the special smooth structure and does not depend on the vertical coordinates, then the Schouten curvature tensor coincides with the Riemannian curvature tensor. The Schouten curvature tensor is used to write the Jacobi equation for the distribution. This leads to studies on second-order optimality conditions for the horizontal geodesics in sub-Riemannian geometry. New example of a distribution with abnormal geodesics is constructed, their optimality being an open problem.



  1. Bliss G. A. Lectures on the calculus of variations. Chicago, Illinois, The University of Chicago Press, 292 p
  2. Vershik A. M., Gershkovich V. Ya. Nonholonomic dynamical systems. Geometry of distributions and variational problems. Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr. VINITI 16 (1987) 5-85
  3. Vershik A. M., Gershkovich V. Ya. The nonholonomic Laplace operator. Probl. Mat. Anal. 11 (1990) 96-108
  4. Gromov M. Carnot-Carathé odory spaces seen from within. Prog. Math. 1996, 144, 79-323
  5. Vranceanu G. Parallé lisme et courbure dans une varié té non holonome. Atti del congresso lnternaz. del Mat. di Bologna, 1928, 6
  6. Schouten J. A., van Kampen D. Zur Einbettung und Krü mmungstheorie nichtholonomer Gebilde. Math. Annalen. 1930. V. 103. P. 752-783
  7. Schouten J. A., Kulk V. D. Pfaffs problem and its generalization. Oxford: Clarendon Press, 1949
  8. Vagner V. V. Differentsialnaya geometriya negolonomnykh mnogoobrazij [The differential geometry of nonholonomic manifolds]. Kazan (1939)
  9. Vagner V. V. Geometricheskaya interpretatsiya dvizheniya negolonomnyh dynamicheskih system [Geometric interpretation of the motion of nonholonomic dynamical systems]. Trudy seminara po vectornomu i tensornomu analisu. 1941. N 5. PP. 301-327.
  10. Gorbatenko E. M. The differential geometry of nonholonomic manifolds according to V. V. Vagner. Geom. Sb. Tomsk univ. 26 (1985) 31-43. (in Russ. )
  11. Krym V. R. Jacobi fields for a nonholonomic distribution. Vestnik St. Petersburg University: Mathematics 43(4) (2010) 232-241
  12. Arutyunov A. V. Second-order conditions in extremal problems. The abnormal points. Transactions of the American Mathematical Society. 1998. Vol. 350, № 11. P. 4341-4365
  13. Arutyunov A. V. Usloviya ekstremuma. Anormalnye i vyrozhdennye zadachi [Extremum conditions. Anormal and degenerated problems]. Moscow, Factorial, 1997. (in Russ. )
  14. Petrov N. N. Existence of abnormal minimizing geodesics in sub-Riemannian geometry. Vestn. St. Petersbg. Univ., Math. 26(3) (1993) 33-38
  15. Petrov N. N. On the shortest sub-Riemannian geodesics. Differ. Equations 30(5) (1994) 705-711
  16. Petrov N. N. A problem of sub-Riemannian geometry. Differ. Equations 31(6) (1995) 911-916
  17. Montgomery R. A survey of singular curves in sub-Riemannian geometry. J. Dynam. Contr. Syst. 1995, N1, P. 49-90
  18. Dmitruk A. V. Quadratic sufficient minimality conditions for abnormal sub-Riemannian geodesics. J. Math. Sci. 2001, V. 104, N1, P. 779-829
  19. Bryant R. L., Hsu L. Rigidity of integral curves of rank-2 distributions. Invent. Math. 1993, V. 114, N2, P. 435-461
  20. Philippov A. F. On some questions of the theory of optimal control. Vestnik Moscovskogo Gosudarstvennogo Univ., ser. Math. and Mech. 1959. N2. P. 25-32
  21. Rashevskii P. K. Any two points of a totally nonholonomic space can be connected by an admissible curve. Uchenye zapiski Moskovskogo ped. in-ta im. Libknechta, ser. phys. -mat. 1938. N2. P. 83-94
  22. Chow W. L. Ű ber Systeme von linearen partiellen Differential-gleichungen erster Ordnung. Math. Ann. 1939. V. 117, N1. P. 98-105
  23. Krym V. R., Petrov N. N. Causal structures on smooth manifolds. Vestn. St. Petersbg. Univ., Math. 34(2) (2001) 1-6
  24. Krym V. R., Petrov N. N. The curvature tensor and the Einstein equations for a four-dimensional nonholonomic distribution. Vestn. St. Petersbg. Univ., Math. 41(3) (2008) 256-265
  25. Krym, V. R. The topological quantization of charges in the Kaluza-Klein theory. Vestn. St. -Petersburg. un-ta, ser. 4. 2009. N 3. PP. 3-12. (In Russ. )
  26. Krym, V. R. Geodesic equations for a charged particle in the unified theory of gravitational and electromagnetic interactions. Theor. Math. Phys. 119(3) (1999) 811-820
  27. Krym, V. R., Petrov, N. N. Equations of motion of a charged particle in a five-dimensional model of the general theory of relativity with a nonholonomic four-dimensional velocity space. Vestn. St. Petersbg. Univ., Math. 40(1) (2007) 52-60
  28. Aharonov Y., Bohm D. Significance of electromagnetic potentials in the quantum theory. Phys. Rev., 2nd series. 1959. Vol. 115. N 3. P. 485-491
  29. Esteban M. J., Georgiev V., Sere E. Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations. Calculus of Variations. 1996. V. 4. P. 265-281
  30. Krym V. R. Index form for nonholonomic distributions. Vestnik St. Petersburg University: Mathematics 45(2) (2012) 73-81
  31. Krym V. R. The Schouten Curvature for a Nonholonomic Distribution in Sub-Riemannian Geometry and Jacobi Fields. Proceedings of the School-Seminar on Optimization Problems and their Applications (OPTA-SCL 2018). Omsk, Russia, July 8-14, 2018. CEUR Workshop Proceedings, v. 2098 (2018).
  32. Matveev N. M. Metody integrirovaniya obyknovennykh differentialnykh uravnenyi [Methods of integration of ordinary differential equations]. Moscow: Higher School, 1963
  33. Hartman P. Ordinary differential equations. 2nd ed., unabridged, corrected republication of the 1982 original edn. Philadelphia, PA: SIAM (2002)
  34. Burago Yu. D., Zalgaller V. A. Vvedenie v rimanovu geometriyu [Introduction in Riemannian geometry]. St. -Petersburg, Nauka, 1994
  35. Zelikin M. I. Optimalnoe upravlenie i variatsionnoe ischislenie [Optimal control and variational calculus]. Moscow, LENAND, 2017

Full text (pdf)