Abstract:
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 $\pi[\omega]$.
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