On an Algebro-Geometric Approach to Ribaucour Transformations and Bianchi Cubes
Matematičeskie zametki, Tome 111 (2022) no. 5, pp. 717-725.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we develop an algebro-geometric approach to Ribaucour transformations and Bianchi cubes of orthogonal nets. Explicit transformations of algebro-geometric data are described that lead to the Ribaucour transformations and Bianchi cubes of algebro-geometric orthogonal nets.
Mots-clés : orthogonal nets, Bianchi cube, Ribaucour transformations.
@article{MZM_2022_111_5_a4,
     author = {E. V. Glukhov},
     title = {On an {Algebro-Geometric} {Approach}  to {Ribaucour} {Transformations} and {Bianchi} {Cubes}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {717--725},
     publisher = {mathdoc},
     volume = {111},
     number = {5},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_111_5_a4/}
}
TY  - JOUR
AU  - E. V. Glukhov
TI  - On an Algebro-Geometric Approach  to Ribaucour Transformations and Bianchi Cubes
JO  - Matematičeskie zametki
PY  - 2022
SP  - 717
EP  - 725
VL  - 111
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2022_111_5_a4/
LA  - ru
ID  - MZM_2022_111_5_a4
ER  - 
%0 Journal Article
%A E. V. Glukhov
%T On an Algebro-Geometric Approach  to Ribaucour Transformations and Bianchi Cubes
%J Matematičeskie zametki
%D 2022
%P 717-725
%V 111
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2022_111_5_a4/
%G ru
%F MZM_2022_111_5_a4
E. V. Glukhov. On an Algebro-Geometric Approach  to Ribaucour Transformations and Bianchi Cubes. Matematičeskie zametki, Tome 111 (2022) no. 5, pp. 717-725. http://geodesic.mathdoc.fr/item/MZM_2022_111_5_a4/

[1] I. M. Krichever, “Algebro-geometricheskie $n$-ortogonalnye krivolineinye sistemy koordinat i resheniya uravnenii assotsiativnosti”, Funkts. analiz i ego pril., 31:1 (1997), 32–50 | DOI | MR | Zbl

[2] D. A. Berdinskii, I. P. Rybnikov, “Ob ortogonalnykh krivolineinykh sistemakh koordinat v prostranstvakh postoyannoi krivizny”, Sib. matem. zhurn., 52:3 (2011), 502–511 | MR | Zbl

[3] E. V. Glukhov, O. I. Mokhov, “Ob algebro-geometricheskikh metodakh postroeniya ploskikh diagonalnykh metrik spetsialnogo vida”, UMN, 74:4 (448) (2019), 185–186 | DOI | MR | Zbl

[4] E. V. Glukhov, O. I. Mokhov, “Ob algebro-geometricheskikh metodakh postroeniya podmnogoobrazii s ploskoi normalnoi svyaznostyu i golonomnoi setyu linii krivizny”, Funkts. analiz i ego pril., 54:3 (2020), 26–37 | DOI | MR | Zbl

[5] A. E. Mironov, I. A. Taimanov, “Ortogonalnye krivolineinye sistemy koordinat, otvechayuschie singulyarnym spektralnym krivym”, Funktsionalnye prostranstva, teoriya priblizhenii, nelineinyi analiz, Tr. MIAN, 255, Nauka, MAIK “Nauka/Interperiodika”, M., 2006, 180–196 | MR

[6] V. Zakharov, “Description of the n-orthogonal curvilinear coordinate systems and Hamiltonian integrable systems of hydrodynamic type. I. Integration of the Laméequations”, Duke Math. J., 94:1 (1998), 103–139 | DOI | MR | Zbl

[7] S. P. Tsarev, “Geometriya gamiltonovykh sistem gidrodinamicheskogo tipa. Obobschennyi metod godografa”, Izv. AN SSSR. Ser. matem., 54:5 (1990), 1048–1068 | MR | Zbl

[8] O. I. Mokhov, E. V. Ferapontov, “O nelokalnykh gamiltonovykh operatorakh gidrodinamicheskogo tipa, svyazannykh s metrikami postoyannoi krivizny”, UMN, 45:3 (273) (1990), 191–192 | MR | Zbl

[9] E. V. Ferapontov, “Differentsialnaya geometriya nelokalnykh gamiltonovykh operatorov gidrodinamicheskogo tipa”, Funkts. analiz i ego pril., 25:3 (1991), 37–49 | MR | Zbl

[10] B. Dubrovin, “Geometry of 2D Topological Field Theories”, Integrable Systems and Quantum Groups, Lecture Notes in Math., 1620, Springer, Berlin, 1996, 120–348 | DOI | MR | Zbl

[11] O. I. Mokhov, “Soglasovannye i pochti soglasovannye metriki”, UMN, 55:4 (334) (2000), 217–218 | DOI | MR | Zbl

[12] O. I. Mokhov, “Pary Laksa dlya uravnenii, opisyvayuschikh soglasovannye nelokalnye skobki Puassona gidrodinamicheskogo tipa, i integriruemye reduktsii uravnenii Lame”, TMF, 138:2 (2004), 283–296 | DOI | MR | Zbl

[13] O. I. Mokhov, “Rimanovy invarianty poluprostykh nelokalno-bigamiltonovykh sistem gidrodinamicheskogo tipa i soglasovannye metriki”, UMN, 65:6 (396) (2010), 189–190 | DOI | MR | Zbl

[14] L. Bianchi, “Le trasformazioni di Ribaucour dei sistemi $n$-pli ortogonali e il teorema generale di permutabilità”, Ann. Mat. Ser. III, 27:1 (1918), 183–257 | DOI | Zbl

[15] M. Dajczer, R. Tojeiro, “An Extension of the Classical Ribaucour Transformation”, Proc. London Math. Soc. (3), 85:1 (2002), 211–232 | DOI | MR | Zbl

[16] U. Hertrich-Jeromin, Introduction to Möbius Differential Geometry, Cambridge Univ. Press, Cambridge, 2003 | MR

[17] E. I. Ganzha, S. P. Tsarev, “Algebraicheskaya formula superpozitsii i polnota preobrazovanii Beklunda $(2+1)$-mernykh integriruemykh sistem”, UMN, 51:6 (312) (1996), 197–198 | DOI | MR | Zbl

[18] F. E. Burstall, U. Hertrich-Jeromin, “The Ribaucour transformation in Lie sphere geometry”, Differential Geom. Appl., 24:5 (2006), 503–520 | DOI | MR | Zbl

[19] L. Florit, M. Dajczer, R. Tojeiro, “The vectorial Ribaucour transformation for submanifolds and applications”, Trans. Amer. Math. Soc., 359:10 (2005), 4977–4997 | MR

[20] A. I. Bobenko, Y. B. Suris, Discrete Differential Geometry: Integrable Structure, Grad. Stud. in Math., 98, Amer. Math. Soc., Providence, RI, 2009 | MR

[21] A. A. Akhmetshin, Yu. S. Volvovskii, I. M. Krichever, “Diskretnye analogi metrik Darbu–Egorova”, Solitony, geometriya, topologiya – na perekrestkakh, Tr. MIAN, 225, Nauka, MAIK “Nauka/Interperiodika”, M., 1999, 21–45 | MR | Zbl