The problem of constructive equivalence in differential geometry

B. M. Dubrov   ; B. P. Komrakov

Sbornik. Mathematics, Tome 191 (2000) no. 5, pp. 655-681

Résumé

The present paper is devoted to the algorithmic construction of diffeomorphisms establishing the equivalence of geometric structures. For structures of finite type the problem reduces to integration of completely integrable distributions with a known symmetry algebra and further to integration of Maurer–Cartan forms. We develop algorithms that reduce this problem to integration of closed 1-forms and equations of Lie type that are characterized by the fact that they admit a non-linear superposition principle. As an application we consider the problem of constructive equivalence for the structures of absolute parallelism and for the transitive Lie algebras of vector fields on manifolds.

Bibliographie

[1] Bernshtein I. N., Rozenfeld I. B., “Odnorodnye prostranstva beskonechnomernykh algebr Li i kharakteristicheskie klassy sloenii”, UMN, 28:4 (1973), 103–138 | MR | Zbl

[2] Vinberg E. B., Onischik A. L., Seminar po gruppam Li i algebraicheskim gruppam, Nauka, M., 1988 | MR

[3] Kamke E., Spravochnik po obyknovennym differentsialnym uravneniyam, Nauka, M., 1971 | MR

[4] Kobayasi Sh., Gruppy preobrazovanii v differentsialnoi geometrii, Nauka, M., 1986 | MR

[5] Kobayasi Sh., Nomidzu T., Osnovy differentsialnoi geometrii, T. 1, Nauka, M., 1969

[6] Losik M. V., “O kharakteristicheskikh klassakh struktur na mnogoobraziyakh”, Funkts. analiz i ego prilozh., 21:3 (1987), 38–52 | MR

[7] Ovsyannikov L. V., Gruppovoi analiz differentsialnykh uravnenii, Nauka, M., 1978 | MR

[8] Olver P., Prilozheniya grupp Li k differentsialnym uravneniyam, Mir, M., 1989 | MR | Zbl

[9] Onischik A. L., “Nekotorye ponyatiya i primeneniya teorii neabelevykh kogomologii”, Tr. MMO, 17, URSS, M., 1967, 45–88 | MR | Zbl

[10] Onischik A. L., Topologiya tranzitivnykh grupp preobrazovanii, Fizmatlit, M., 1995 | MR | Zbl

[11] Sternberg Sh., Lektsii po differentsialnoi geometrii, Mir, M., 1989

[12] Fuks D. B., Kogomologii beskonechnomernykh algebr Li, Nauka, M., 1984 | MR | Zbl

[13] Duzhin S. V., Lychagin V. V., “Symmetries of distributions and quadrature of ordinary differential equations”, Acta Appl. Math., 24 (1991), 25–37 | MR

[14] Gardner R. B., The method of equivalence and its applications, SIAM, Philadelphia, 1989 | MR

[15] Griffiths P., “On Cartan method of Lie groups and moving frames as applied to existence and uniqueness questions in differential geometry”, Duke Math. J., 41 (1974), 775–814 | DOI | MR | Zbl

[16] Hermann R., “Equivalence invariants for submanifolds of homogeneous spaces”, Math. Ann., 158 (1965), 284–289 | DOI | MR | Zbl

[17] Hermann R., Lie groups: History, frontiers and applications. Vol. I. Sophus Lie's 1880 transformation group paper, Math. Sci. Press, Brookline, 1975 | MR

[18] Komrakov B. P., Lychagin V. V., Symmetries and integrals, Preprint No 15, Univ. Oslo, 1993

[19] Lie S., “Allgemeine Untersuchung über Differentialgleichungen, die eine kontinuierliche, endliche Gruppe gestatten”, Math. Ann., 25 (1) (1885), 71–151 | DOI | MR

[20] Lie S., Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen, Teubner, Leipzig, 1891 | Zbl

[21] Lie S., Vorlesungen über continuierlishe Gruppen, Teubner, Leipzig, 1893

[22] Olver P., Symmetry, invariants, and equivalence, Springer-Verlag, New York, 1995 | Zbl

[23] Shnider S., Winternitz P., “Nonlinear equations with superposition principles and the theory of transitive primitive Lie algebras”, Lett. Math. Phys., 8 (1984), 69–78 | DOI | MR | Zbl

[24] Sulanke R., “On E. Cartan's method of moving frame”, Colloq. Math. Soc. János Bolyai, 31 (1979), 681–704 | MR