Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 092, 41 pages      arXiv:0909.4728
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Existence and Construction of Vessiot Connections

Dirk Fesser a and Werner M. Seiler b
a) IWR, Universität Heidelberg, INF 368, 69120 Heidelberg, Germany
b) AG ''Computational Mathematics'', Universität Kassel, 34132 Kassel, Germany

Received May 05, 2009, in final form September 14, 2009; Published online September 25, 2009

A rigorous formulation of Vessiot's vector field approach to the analysis of general systems of partial differential equations is provided. It is shown that this approach is equivalent to the formal theory of differential equations and that it can be carried through if, and only if, the given system is involutive. As a by-product, we provide a novel characterisation of transversal integral elements via the contact map.

Key words: formal integrability; integral element; involution; partial differential equation; Vessiot connection; Vessiot distribution.

pdf (533 kb)   ps (414 kb)   tex (80 kb)


  1. Anderson I., Kamran N., Olver P.J., Internal, external, and generalized symmetries, Adv. Math. 100 (1993), 53-100.
  2. Arnold V.I., Geometrical methods in the theory of ordinary differential equations, Grundlehren der Mathematischen Wissenschaften, Vol. 250, Springer-Verlag, New York, 1988.
  3. Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A., Exterior differential systems, Mathematical Sciences Research Institute Publications, Vol. 18, Springer-Verlag, New York, 1991.
  4. Cartan É., Les Systèmes différentiels extérieurs et leurs applications géométriques, Actualités Sci. Ind., no. 994, Hermann et Cie., Paris, 1945.
  5. Fackerell E.D., Isovectors and prolongation structures by vessiot's vector field formulation of partial differential equations, in Geometric Aspects of the Einstein Equations and Integrable Systems (Scheveningen, 1984), Editor R. Martini, Lecture Notes in Phys., Vol. 239, Springer-Verlag, Berlin, 1985, 303-321.
  6. Fesser D., On Vessiot's theory of partial differential equations. A geometric approach to constructing infinitesimal solutions, PhD Thesis, Saarbrücken, 2008.
  7. Fesser D., Seiler W.M., Vessiot connections of partial differential equations, in Global Integrability of Field Theories - GIFT 2006, Editors J. Calmet, W.M. Seiler and R.W. Tucker, Universitätsverlag Karlsruhe. Karlsruhe, 2006, 111-134.
  8. Globke W., Eine Objektorientierte Programmierumgebung für Differentialgeometrische Berechnungen in MuPAD, Project Thesis, Fakultät für Informatik, Universität Karlsruhe, 2006.
  9. Golubitsky M., Guillemin V.W., Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York - Heidelberg, 1973.
  10. Hartley D.H., Tucker R.W., A constructive implementation of the Cartan-Kähler theory of exterior differential systems, J. Symbolic Comput. 12 (1991), 655-667.
  11. Hausdorf M., Seiler W.M., An efficient algebraic algorithm for the geometric completion to involution, Appl. Algebra Engrg. Comm. Comput. 13 (2002), 163-207.
  12. Ivey T.A., Landsberg J.M., Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, Vol. 61, American Mathematical Society, Providence, RI, 2003.
  13. Krasilshchik I.S., Lychagin V.V., Vinogradov A.M., Geometry of jet spaces and nonlinear partial differential equations, Advanced Studies in Contemporary Mathematics, Vol. 1, Gordon and Breach Science Publishers, New York, 1986.
  14. Kruglikov B.S., Lychagin V.V., Geometry of differential equations, in Handbook of Global Analysis, Editors D. Krupka and D. Saunders, Elsevier, Amsterdam, 2008, 725-771.
  15. Lychagin V.V., Homogeneous geometric structures and homogeneous differential equations, in The Interplay between Differential Geometry and Differential Equations, Editor V.V. Lychagin, Amer. Math. Soc. Transl. Ser. 2, Vol. 167, Amer. Math. Soc., Providence, RI, 1995, 143-164.
  16. Lychagin V.V., Singularities of multivalued solutions of nonlinear differential equations, and nonlinear phenomena, Acta Appl. Math. 3 (1985), 135-173.
  17. Modugno M., Covariant quantum mechanics, Unpublished manuscript, Dept. of Mathematics, University of Florence, 1999.
  18. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1986.
  19. Pommaret J.F., Systems of partial differential equations and Lie pseudogroups, Mathematics and its Applications, Vol. 14, Gordon & Breach Science Publishers, New York, 1978.
  20. Saunders D.J., The geometry of jet bundles, London Mathematical Society Lecture Note Series, Vol. 142, Cambridge University Press, Cambridge, 1989.
  21. Seiler W.M., Completion to involution and semi-discretisations, Appl. Numer. Math. 42 (2002), 437-451.
  22. Seiler W.M., Spencer cohomology, differential equations, and Pommaret bases, in Gröbner Bases and Symbolic Analysis, Editors M. Rosenkranz and D.M. Wang, Radon Ser. Comput. Appl. Math., Vol. 2, Walter de Gruyter, Berlin, 2007, 169-216.
  23. Seiler W.M., Involution - the formal theory of differential equations and its applications in computer algebra, Algorithms and Computation in Mathematics, Vol. 24, Springer-Verlag, Berlin - Heidelberg, 2009.
  24. Stormark O., Lie's structural approach to PDE systems, Encyclopedia of Mathematics and its Applications, Vol. 80, Cambridge University Press, Cambridge, 2000.
  25. Tuomela J., On singular points of quasilinear differential and differential-algebraic equations, BIT 37 (1997), 968-977.
  26. Tuomela J., On the resolution of singularities of ordinary differential equations, Numer. Algorithms 19 (1998), 247-259.
  27. Vassiliou P.J., Vessiot structure for manifolds of (p,q)-hyperbolic type: Darboux integrability and symmetry, Trans. Amer. Math. Soc. 353 (2001), 1705-1739.
  28. Vessiot E., Sur une théorie nouvelle des problèmes généraux d'intégration, Bull. Soc. Math. France 52 (1924), 336-395.

Previous article   Next article   Contents of Volume 5 (2009)