Asymptotic lines on the pseudo-spherical surfaces

A. V. Kostin

Vladikavkazskij matematičeskij žurnal, Tome 21 (2019) no. 1, pp. 16-26

Résumé

Consider the three-dimensional extended Lobachevsky space. In a proper area of Lobachevsky space take the ‘complete’ pseudosphere, that is, a surface of rotation of a straight line around a given parallel straight line. One part of it is embedded into Euclidean space in the form of the Beltrami–Minding funnel, the other one into three-dimensional Minkowski space as an analogue of the pseudosphere in this space. The interpretations of imaginary asymptotic lines on this pseudospherical surface with the Lobachevsky metric in Minkowski space are considered. Imaginary asymptotic lines on the pseudo-Euclidean continuation of the pseudosphere can be interpreted as real asymptotic lines on the surface of constant curvature with indefinite metric. These surfaces are other pseudo-Euclidean analogs of the Beltrami–Minding pseudosphere. The properties of the asymptotic lines on the pseudospheres with de Sitter metric in the three-dimensional Minkowsky space are studied. The considered properties of asymptotic lines on pseudospheres of pseudo-Euclidean space (Minkowski space) are similar to that of asymptotic lines on the Beltrami–Minding pseudosphere in Euclidean space. Areas of quadrangles of the asymptotic net on a surface of constant negative curvature in Euclidean space can be found by the Hazzidakis formula. These results are transferred to surfaces of constant curvature with indefinite metric in Minkowski space.