Canonical $A$-deformations preserving the lengths of lines of curvature on a surface

L. L. Beskorovainaya

Sbornik. Mathematics, Tome 26 (1975) no. 2, pp. 151-164

Résumé

In this paper, infinitesimal deformations which preserve the area element of a surface in $E_3$ ($A$-deformations) which also preserve the lengths of lines of curvature are studied. Here $A$-deformations are considered up to infinitesimal bendings (which constitute the trivial case for the problem posed). Such $A$-deformations are also called canonical. For regular surfaces of nonzero total curvature (without umbilic points) the problem indicated reduces to a homogeneous second order partial differential equation of elliptic type. In this paper a series of results about the existence and arbitrariness of canonical $A$-deformations is obtained. The basic results are valid for surfaces in the large. Bibliography: 20 titles.

Bibliographie

[1] N. V. Efimov, “Kachestvennye voprosy teorii deformatsii poverkhnosti”, Uspekhi matem. nauk, 1948, no. 2(24), 45–158 | MR

[2] I. N. Vekua, Obobschennye analiticheskie funktsii, Fizmatgiz, Moskva, 1959 | MR

[3] I. N. Vekua, Novye metody resheniya ellipticheskikh uravnenii, Gostekhizdat, Moskva–Leningrad, 1948 | MR

[4] I. N. Vekua, “Sistemy differentsialnykh uravnenii pervogo poryadka ellipticheskogo tipa i granichnye zadachi s primeneniem k teorii obolochek”, Matem. sb., 31(73) (1952), 217–314 | MR | Zbl

[5] I. N. Vekua, “Integrirovanie uravnenii sfericheskoi obolochki”, Prikl. matem. mekh., IX:5 (1945), 368–388

[6] A. V. Bitsadze, Kraevye zadachi dlya ellipticheskikh uravnenii vtorogo poryadka, izd-vo «Nauka», Moskva, 1966

[7] K. Miranda, Uravneniya s chastnymi proizvodnymi ellipticheskogo tipa, IL, Moskva, 1957

[8] E. T. Uitteker, G. N. Vatson, Kurs sovremennogo analiza, t. II, Fizmatgiz, Moskva, 1963

[9] V. F. Kagan, Osnovy teorii poverkhnostei, Gostekhizdat, Moskva–Leningrad, 1947 ; 1948 | MR

[10] A. V. Pogorelov, Vneshnyaya geometriya vypuklykh poverkhnostei, izd-vo «Nauka», Moskva, 1969 | MR

[11] Roger Boudet, Sur quelques propriétés géométriques des transformations infinitésimales des surfaces, Thèse, doct. sci. math., Fac. sci. Univ. Aix-Marseille, 1961, 78 | MR | Zbl

[12] Paul Vincensini, “Sur les déformations équivalentes infinitésimales des surfaces”, Rev. Univ. nac. Tucumán, A14:1–2 (1962), 177–188 | MR

[13] Rosca Radu, “Sur les congruences dont les nappes focales sont en correspondance pseudo-isométrique”, Rev. roumaine math. pures et apl., 12:5 (1967), 697–708 | MR | Zbl

[14] P. G. Kolobov, “Ekviafinnaya beskonechno malaya deformatsiya poverkhnosti”, Uchenye zapiski Kabardino-Balkarsk. un-ta, seriya fiz.-matem., no. 24, 1965, 54–56 | MR

[15] P. G. Kolobov, “O beskonechno malykh deformatsiyakh poverkhnosti s sokhraneniem ploschadi”, Uchenye zapiski Kabardino-Balkarsk. un-ta, seriya fiz.-matem., no. 30, 1966, 65–68 | MR

[16] P. G. Kolobov, I. V. Vasilego, “O beskonechno malykh ekviarealnykh deformatsiyakh poverkhnosti”, Matematika, nekotorye ee prilozheniya i metodika prepodavaniya, Rostov, 1972, 15–19

[17] V. A. Tikhonov, “O beskonechno malykh $p$-izgibaniyakh”, Izv. VUZov, Matematika, 1971, no. 7, 94–98 | Zbl

[18] V. P. Bilousova, P. L. Simokin, Z. A. Mikheeva, “Neskinchenno mali afinni deformatsiïpoverkhon”, Visnik KDU, seriya matem. ta mekh., 1969, no. 1, 25–41 | MR

[19] L. L. Bezkorovaina, “Pro neskinchenno mali deformatsiïpoverkhon”, Druga naukova konferentsiya molodikh matem. Ukraïni, izd.-vo «Naukova dumka», Kiïv, 1966, 39–42

[20] L. L. Bezkorovaina, “Pro neskinchenno mali deformatsiï, scho zberigayut dovzhinu asimptotichnikh linii”, Materiali univer. naukovoïkonferentsiïmolodikh vchenikh, prisvyachenoï100-richchyu z dnya narodzhennya V. I. Lenina (prirodnichi nauki), Odesa, 1970, 104–109