Isometric immersion in \(E^ 3\) of a convex domain of the Lobachevskij plane containing two horocycles (Q1086850)

The paper is devoted to the investigation of the possibility of a regular isometric immersion of unbounded convex domains of the Lobachevskij plane into 3-dimensional Euclidean space \(E^ 3\). The basic results in this area are due to Hilbert (the Lobachevskij plane does not admit a regular isometric immersion in \(E^ 3)\), Efimov (the Lobachevskij half-plane does not admit a regular isometric immersion in \(E^ 3)\) and Poznjak (any convex polygon with a finite number of infinitely distant vertices and null angles at them can be realized in \(E^ 3\) as a regular surface). In the present paper the authors prove two theorems: 1) An expanding strip of the Lobachevskij plane can be realized in \(E^ 3\) as a regular \((C^ 3\)-smooth) surface. 2) Any two horocycles of the Lobachevskij plane can be included in a convex domain which can be immersed regularly and isometrically in \(E^ 3\).