Geodesic
Infinitesimal higher order bendings of multidimensional surfaces in spaces of constant curvature
Sbornik. Mathematics, Tome 61 (1988) no. 1, pp. 65-85 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Infinitesimal bendings of order $r\geqslant1$ are considered, including analytic bendings ($r=\infty$), of an $n$-dimensional surface $F$ in an $m$-dimensional ($1\leqslant n) space $W$ of constant curvature. It is proved that to any solution of an $r$ times formally varied system of Gauss–Codazzi–Ricci equations there corresponds an infinitesimal bending of order $r$ of the surface $F$ in $W$. A general form is established for solutions of this system that determine infinitesimal motions of various orders. By using these results we obtain criteria for rigidity and nonrigidity of order $r\leqslant1$, and also for analytic bendability and nonbendability of a class of multidimensional surfaces of codimension $p\geqslant1$ in flat spaces, which contains, in particular, Riemannian products of hypersurfaces. Bibliography. 13 titles. 
@article{SM_1988_61_1_a4,
     author = {P. E. Markov},
     title = {Infinitesimal higher order bendings of multidimensional surfaces in spaces of constant curvature},
     journal = {Sbornik. Mathematics},
     pages = {65--85},
     year = {1988},
     volume = {61},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1988_61_1_a4/}
}
TY  - JOUR
AU  - P. E. Markov
TI  - Infinitesimal higher order bendings of multidimensional surfaces in spaces of constant curvature
JO  - Sbornik. Mathematics
PY  - 1988
SP  - 65
EP  - 85
VL  - 61
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1988_61_1_a4/
LA  - en
ID  - SM_1988_61_1_a4
ER  - 
%0 Journal Article
%A P. E. Markov
%T Infinitesimal higher order bendings of multidimensional surfaces in spaces of constant curvature
%J Sbornik. Mathematics
%D 1988
%P 65-85
%V 61
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1988_61_1_a4/
%G en
%F SM_1988_61_1_a4
P. E. Markov. Infinitesimal higher order bendings of multidimensional surfaces in spaces of constant curvature. Sbornik. Mathematics, Tome 61 (1988) no. 1, pp. 65-85. http://geodesic.mathdoc.fr/item/SM_1988_61_1_a4/

[1] Efimov N. V., “Kachestvennye voprosy teorii deformatsii poverkhnostei”, UMN, 3 (1948), 47–158 | MR | Zbl

[2] Efimov N. V., “Nekotorye predlozheniya o zhestkosti i neizgibaemosti”, UMN, 7:5 (1952), 215–224 | MR | Zbl

[3] Markov P. E., “Beskonechno malye izgibaniya nekotorykh mnogomernykh poverkhnostei”, Matem. zametki., 27:3 (1980), 469–479 | MR | Zbl

[4] Markov P. E., “Beskonechno malye izgibaniya odnogo klassa mnogomernykh poverkhnostei s kraem”, Matem. sb., 12(54) (1983), 48–59

[5] Chernyavskaya I. A., “Beskonechno malye izgibaniya pervogo i vtorogo poryadkov poverkhnostei v prostranstve Lobachevskogo”, Comment. Math. Univers. Carolinae, 16:3 (1975), 399–424 | MR

[6] Pogorelov A. V., Vneshnyaya geometriya vypuklykh poverkhnostei, Nauka, M., 1969 | MR

[7] Markov P. E., “Beskonechno malye izgibaniya vysshikh poryadkov mnogomernykh poverkhnostei”, Ukr. geom. sb., 25 (1982), 87–94 | MR | Zbl

[8] Eizenkhart L. P., Rimanova geometriya, GIIL, M., 1948

[9] Kartan E., Geometriya rimanovykh prostranstv, ONTI NKTP SSSR, M.-L., 1936

[10] Markov P. E., Beskonechno malye izgibaniya $n$-mernykh poverkhnostei v $m$-mernykh prostranstvakh postoyannoi krivizny, Dis. $\dots$ kand. fiz.-matem. nauk, Minsk, 1978

[11] Klimentov S. B., “O prodolzhenii beskonechno malykh izgibanii vysshikh poryadkov odnosvyaznoi poverkhnosti polozhitelnoi krivizny”, Matem. zametki., 36:3 (1984), 393–403 | MR | Zbl

[12] Mur Dzh., “Izometricheskie pogruzheniya rimanovykh proizvedenii”, Issledovaniya po metricheskoi teorii poverkhnostei, Mir, M., 1980, 254–276

[13] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974 | MR