Some remarks on the tubes of negative curvature

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.