Isometric immersions of domains of Lobachevsky space in Euclidean spaces

Immersions of domains of the $n$-dimensional Lobachevsky space $L^n$ in the $(2n-1)$-dimensional Euclidean space $E^{2n-1}$ are studied. It is shown that the problem of isometric immersion of domains of $L^n$ in $E^{2n-1}$ is reduced to the study of a certain system of nonlinear partial differential equations, yielding the sine-Gordon equation as one of the special cases.