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A contravariant theory of differential prolongation in models of spaces with connection

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Abstract

This is a survey of a series of the author's studies on the geometry of differential-geometric structures, with applications to the geometry of differential equations.

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Literature cited

  1. V. I. Bliznikas, “On the theory of curvature of spaces of supporting elements,” Lit. Mat. Sb.,5, No. 1, 9–24 (1965).

    Google Scholar 

  2. V. I. Bliznikas and Z. Yu. Lupeikis, “Geometry of differential equations,” in: Algebra, Topology, Geometry [in Russian], Vol. 11 (Itogi Nauki i Tekh. VINITI Akad. Nauk SSSR), Moscow (1974), pp. 209–259.

    Google Scholar 

  3. V. I. Bliznikas and Z. Yu. Lupeikis, “On the geometry of certain systems of partial differential equations,” in: Proceedings of the Geometry Seminar [in Russian], Vol. 5, VINITI (1974), pp. 135–168.

  4. A. M. Vasil'ev, “Differential algebra. Contravariant analytic methods in differential geometry,” in: Problems in Geometry [in Russian], Vol. 10 (Itogi Nauki i Tekh. VINITI Akad. Nauk SSSR), Moscow (1978), pp. 5–24.

    Google Scholar 

  5. A. M. Vinogradov, “Geometry on nonlinear differential equations,” in: Problems in Geometry [in Russian], Vol. 11 (Itogi Nauki i Tekh. VINITI Akad. Nauk SSSR), Moscow (1980), pp. 89–134.

    Google Scholar 

  6. R. V. Vosilyus, “Geometry of fibered submanifolds,” in: Proceedings of the Geometry Seminar [in Russian], Vol. 5, VINITI (1974), pp. 201–237.

  7. R. V. Vosilyus, “Formal differentiation in spaces of geometric objects,” Liet. Mat. Rinkinys (Lit. Mat. Sb.),15, No. 4, 17–40 (1976).

    Google Scholar 

  8. R. V. Vosilyus, “Generalized linear structures and the geometry of submanifolds. Part I,” Liet. Mat. Rinkinys (Lit. Mat. Sb.),17, No. 4, 31–82 (1977).

    Google Scholar 

  9. R. V. Vosilyus, “The geometry of systems of differential equations. I. Formalization of of the Frölicher-Nijenhuis theory,” Liet. Mat. Rinkinys (Lit. Mat. Sb.),21, No. 2, 67–80 (1981).

    Google Scholar 

  10. R. V. Vosilyus, “The geometry of systems of differential equations. II. Contravariant theory of differential prolongation,” Liet. Mat. Rinkinys,21, No. 4, 19–34 (1981).

    Google Scholar 

  11. R. V. Vosilyus, “The geometry of systems of differential equations. III. Intrinsic formalization of differential-geometric structures,” Liet. Mat. Rinkinys (Lit. Mat. Sb.),22, No. 1, 40–54 (1982).

    Google Scholar 

  12. R. V. Vosilyus, “The geometry of systems of differential equations. IV. The determining Lie-Bäcklund system,” Liet. Mat. Rinkinys (Lit. Mat. Sb.),22, No. 2, 28–43 (1982).

    Google Scholar 

  13. R. V. Vosilyus, “The geometry of systems of differential equations. V. The determining Lie structure,” Liet Mat. Rinkinys (Lit. Mat. Sb.),22, No. 3, 40–54 (1982).

    Google Scholar 

  14. L. E. Evtushik, “Differential connections and infinitesimal transformations of the prolonged pseudogroup,” in: Proceedings of the Geometry Seminar [in Russian], Vol. 2, VINITI (1969), pp. 119–150.

  15. N. Kh. Ibragimov and R. L. Anderson, “Groups of contact Lie-Bäcklund transformations,” Dokl. Akad. Nauk SSSR,227, No. 3, 539–542 (1976).

    Google Scholar 

  16. B. A. Kupershmidt, “On the geometry of jet manifolds,” Usp. Mat. Nauk,30, No. 5, 211–212 (1975).

    Google Scholar 

  17. G. F. Laptev, “Distributions of tangent elements,” in: Proceedings of the Geometry Seminar [in Russian], Vol. 3, VINITI (1971), pp. 29–48.

  18. V. V. Lychagin, “Local classification of nonlinear partial differential equations of first order,” Usp. Mat. Nauk,30, No. 1, 101–171 (1975).

    Google Scholar 

  19. Yu. I. Manin, “Algebraic aspects of nonlinear differential equations,” in: Contemporary Problems in Mathematics [in Russian] (Itogi Nauki i Tekh. VINITI Akad. Nauk SSSR), Vol. 11, Moscow (1978), pp. 5–152.

    Google Scholar 

  20. L. V. Ovsyannikov, Group Analysis of Differential Equations, Academic Press, New York (1983).

    Google Scholar 

  21. N. M. Ostianu, “Step bundles,” in: Proceedings of the Geometry Seminar [in Russian], Vol. 5, VINITI (1974), pp. 259–309.

  22. P. A. Shirokov and A. P. Shirokov, Affine Differential Geometry [in Russian], Fizmatgiz, Moscow (1959).

    Google Scholar 

  23. H. Goldschmidt, “Existence theorems for analytic linear partial differential equations,” Ann. Math.,86, No. 2, 246–270 (1967).

    Google Scholar 

  24. H. Goldschmidt, “Integrability criteria for systems of nonlinear partial differential equations,” J. Diff. Geom.,1, No. 3, 269–307 (1967).

    Google Scholar 

  25. V. W. Guillemin and S. Sternberg, “The Lewy counterexample and the local equivalence problem for G-structures,” J. Diff. Geom.,1, No. 2, 127–131 (1967).

    Google Scholar 

  26. A. Kumpera, “Invariants differentiels d'un pseudogroupe de Lie,” J. Diff. Geom.,10, No. 2, 289–345 (1975).

    Google Scholar 

  27. A. Kumpera and D. Spencer, Lie Equations, Vol. I, General Theory, Ann. Math. Studies, No. 73, Princeton Univ. Press, Princeton, N.J. (1972).

    Google Scholar 

  28. M. Kuranishi, Lectures on Involutive Systems of Partial Differential Equations, Publ. Soc. Mat. Sao Paulo (1967).

  29. H. Lewy, “An example of a smooth linear partial differential equation without solutions,” Ann. Math.,66, No. 1, 155–158 (1957).

    Google Scholar 

  30. B. Malgrange, “Equations de Lie. I,” J. Diff. Geom.,6, No. 4, 503–522 (1972).

    Google Scholar 

  31. B. Malgrange, “Equations de Lie. II,” J. Diff. Geom.,7, Nos. 1–2, 117–141 (1972).

    Google Scholar 

  32. Ngô Van Quê, “Du prolongement des espaces fibrés et des structures infinitésimales,” Ann. Inst. Fourier,17, No. 1, 157–223 (1967).

    Google Scholar 

  33. J. Vanžura, “Invariants of submanifolds,” Czechosl. Mat. J.,19, No. 3, 452–468 (1969).

    Google Scholar 

  34. J. Vanžura, “Tensor-invariants of submanifolds,” Czechosl. Mat. J.,21, No. 3, 437–448 (1971).

    Google Scholar 

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Authors
  1. R. V. Vosilyus
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Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 14, pp. 101–176, 1983.

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Vosilyus, R.V. A contravariant theory of differential prolongation in models of spaces with connection. J Math Sci 28, 208–256 (1985). https://doi.org/10.1007/BF02105212

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