Geodesic
Bäcklund maps and Lie–Bäcklund transformations as differential-geometric structures
Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 1, pp. 135-150 
A. K. Rybnikov. Bäcklund maps and Lie–Bäcklund transformations as differential-geometric structures. Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 1, pp. 135-150. http://geodesic.mathdoc.fr/item/FPM_2010_16_1_a10/
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Voir la notice de l'article provenant de la source Math-Net.Ru

This paper is an exposition of the author's report prepared for the International Conference devoted to the centennial anniversary of G. F. Laptev (Laptev seminar – 2009). In the first section, we consider Bäcklund transformations of second-order partial differential equations. In the present work, the theory of Bäcklund transformations is treated as a special branch of the theory of connections. The second section is devoted to differential-geometric structures generated by so-called Lie–Bäcklund transformations (or, equivalently, contact transformations of higher order) that are a special case of diffeomorphisms between the manifolds of holonomic jets. Recall that it was G. F. Laptev who pointed out the possibility of considering differentiable mappings as differential-geometric structures.

[1] Vasilev A. M., Teoriya differentsialno-geometricheskikh struktur., Izd-vo Mosk. un-ta, M., 1987 | MR | Zbl

[2] Evtushik L. E., Lumiste Yu. G., Ostianu N. M., Shirokov A.P., “Differentsialno-geometricheskie struktury na mnogoobraziyakh”, Itogi nauki i tekhn. Ser. Probl. geom. Tr. geom. sem., 9, 1979, 5–246 | MR | Zbl

[3] Ibragimov N. Kh., Gruppy preobrazovanii v matematicheskoi fizike., Nauka, M., 1983 | MR

[4] Laptev G. F., “Osnovnye infinitezimalnye struktury vysshikh poryadkov na gladkom mnogoobrazii”, Itogi nauki i tekhn. Ser. Probl. geom. Tr. geom. sem., 1, 1966, 139–189 | MR | Zbl

[5] Laptev G. F., “K invariantnoi teorii differentsiruemykh otobrazhenii”, Itogi nauki i tekhn. Ser. Probl. geom. Tr. geom. sem., 6, 1974, 37–42 | MR | Zbl

[6] Polyanin L. D., Zaitsev V. F., Zhurov A. I., Metody resheniya nelineinykh uravnenii matematicheskoi fiziki i mekhaniki, Fizmatlit, M., 2003

[7] Rybnikov A. K., “O spetsialnykh svyaznostyakh, opredelyayuschikh predstavlenie nulevoi krivizny dlya evolyutsionnykh uravnenii vtorogo poryadka”, Izv. vyssh. uchebn. zaved. Matematika, 1999, no. 9, 32–41 | MR | Zbl

[8] Rybnikov A. K., “Teoriya svyaznostei i preobrazovaniya Beklunda dlya obschikh differentsialnykh uravnenii s chastnymi proizvodnymi vtorogo poryadka”, Dokl. RAN, 405:1 (2005), 26–29 | MR | Zbl

[9] Rybnikov A. K., “Teoriya svyaznostei i problema suschestvovaniya preobrazovanii Beklunda dlya evolyutsionnykh uravnenii vtorogo poryadka”, Dokl. RAN, 400:3 (2005), 319–322 | MR

[10] Rybnikov A. K., “Teoriya svyaznostei, preobrazovaniya Koula–Khopfa i potentsialy differentsialnykh uravnenii s chastnymi proizvodnymi vtorogo poryadka”, Izv. vyssh. uchebn. zaved. Matematika, 2007, no. 9, 50–70 | MR | Zbl

[11] Rybnikov A. K., “Differentsialno-geometricheskie struktury, opredelyayuschie preobrazovaniya Li–Beklunda”, Dokl. RAN, 425:1 (2009), 25–30 | MR | Zbl

[12] Rybnikov A. K., Semënov K. V., “Svyaznosti Beklunda i otobrazheniya Beklunda, sootvetstvuyuschie evolyutsionnym uravneniyam vtorogo poryadka”, Izv. vyssh. uchebn. zaved. Matematika, 2004, no. 5, 52–68 | MR | Zbl

[13] Forsyth A. K., Theory of Differential Equations, Pt. 4. Vol. 6, Cambridge Univ. Press, Cambridge, 1906

[14] Pirani F. A. E., Robinson D. C., “Sur la définition des transformations de Bäcklund”, C. R. Acad. Sci. Paris Sér. A, 285 (1977), 581–583 | MR | Zbl