On the regular isometric immersion in $E^3$ of unbounded domains of

D. V. Tunitsky

Sbornik. Mathematics, Tome 62 (1989) no. 1, pp. 121-138

Résumé

A wide class of unbounded domains is considered in complete Riemannian manifolds of negative curvature that are homeomorphic to planes, and the possibility of immersing them regularly and isometrically in three-dimensional Euclidean space $E^3$ is investigated. Let a metric of the manifold under consideration be given on the parameter plane $xOy$ by a line element of the form $ds^2=dx^2+B^2(x,y)dy^2$, where $B\in C^4(R^2)$. The set $\pi[\omega]=\{(x,y)\in R^2:|x|\omega(y)\}$ is considered, where $\omega(y)>0$ and is twice continuously differentiable. Let $\pi^*[\omega]$ denote the corresponding domain on the manifold. Then the domain $\pi^*[\omega]$ can be isometrically immersed in $E^3$ by means of a surface of class $C^3$. This result is proved by constructing a smooth solution of a special form of the Gauss–Peterson–Codazzi system of equations in the domain $\pi[\omega]$. Figures: 2. Bibliography: 11 titles.