Geodesic
Non-existence of closed asymptotic curves on the tubes of negative Gaussian curvature with one-to-one spherical mapping
Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 4, pp. 953-977 
G. A. Kovaleva. Non-existence of closed asymptotic curves on the tubes of negative Gaussian curvature with one-to-one spherical mapping. Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 4, pp. 953-977. http://geodesic.mathdoc.fr/item/FPM_1995_1_4_a7/
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Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that on the tubes having negative Gaussian curvature and biunique spherical image there is no closed asymptotic curves and by that a Nirenberg's problem is solved which he has posed in relation with the study of rigidity of T-surfaces.

[1] Nirenberg L., “Rigidity of a class of closed surfaces”, Nonlinear problems, The University of Wisconsin Press, Madison, 1963 | MR

[2] Kovalëva G. A., “O zamknutykh asimptoticheskikh liniyakh na trubkakh otritsatelnoi gaussovoi krivizny”, Tezisy 2-go Vsesoyuznogo simpoziuma po geometrii v tselom, 1967; “Пример гомотопной трубке поверхности с замкнутой асимптотической линией”, Матем. заметки, 3:4 (1968), 403–413, Петрозаводск | MR | Zbl

[3] Kantor B. E., “Kontinuum zamknutykh asimptoticheskikh na trubke otritsatelnoi krivizny”, Zadachi geometrii v tselom dlya pogruzhennykh mnogoobrazii, Mezhvuzovskii sbornik nauchnykh trudov, Rossiiskii gosud. pedagogicheskii universitet im. A. I. Gertsena, S.-Peterburg, 1991, 15–19 | MR

[4] Fenchel W., “On the differential geometry of closed space curves”, Bull. Amer. Math. Soc., 57:1 (1951), 44–54 | DOI | MR | Zbl

[5] Aminov Yu. A., Differentsialnaya geometriya i topologiya krivykh, Nauka, M., 1987 | MR | Zbl

[6] Bukhgolts N. N., Osnovnoi kurs teoreticheskoi mekhaniki, M., 1967