Geodesic
Some remarks on the tubes of negative curvature
Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 4, pp. 1033-1043
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In 1963 L. Nirenberg has showed that the rigidity of a so-called $T$-surface depends on the nonexistence of two closed asymptotic lines on the tubes of negative curvature. In the article we give some conditions sufficient for nonexistence of closed asymptotic curves and besides we formulate and comment a number of problems concerning the exterior geometric structure of the tubes of negative curvature. 
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I. Kh. Sabitov. Some remarks on the tubes of negative curvature. Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 4, pp. 1033-1043. http://geodesic.mathdoc.fr/item/FPM_1995_1_4_a12/

[1] A. D. Aleksandrov, “Ob odnom klasse zamknutykh poverkhnostei”, Mat. sbornik, 4:1 (1938), 69–77

[2] L. Nirenberg, “Rigidity of a class of closed surfaces”, Non linear problems. University of Visconsin Press, 1963, 177–193 | MR | Zbl

[3] M. Shiffman, “On surfaces of stationary area bounded by two circles, or convex curves, in parallel planes”, Ann. of Math., 63:2 (1956), 77–90 | DOI | MR | Zbl

[4] D. Gilbert, S. Kon-Fossen, Naglyadnaya geometriya, Fizmatgiz, 1981, 344 pp. | MR

[5] I. Kh. Sabitov, “Minimalnaya poverkhnost kak diagramma vraschenii sfery”, Matem. zametki, 2:6 (1967), 645–656 | MR