Isometric immersions of domains of $n$-dimensional Lobachevsky space in $(2n-1)$-dimensional Euclidean space

This paper considers regular submanifolds of Euclidean space $E^N$. It is shown that if $R^m$ is a submanifold of negative curvature of $E^N$ and at each point there are $m$ principal directions, then there are hypersurfaces of $R^m$ orthogonal to them; these can be taken as the coordinate hypersurfaces. Furthermore, general properties of an isometric immersion of $n$-dimensional Lobachevsky space $L^n$ in $E^{2n-1}$ are considered. It is proved that, for any $k$-dimensional submanifold of $L^n\subset E^{2n-1}$ for $k\geqslant2$ and $n>2$, the $k$-dimensional volume of its image in $G_{n-1,2n-1}$ under a Grassmann mapping of $L^n$ is greater than the volume of its inverse image. The curvature $\overline K$ of $G_{n-1,2n-1}$ for elements of area tangent to the Grassmann image of $L^n$ lie in the open interval $(0,1)$. A formula is obtained for the curvature of a Grassmann manifold for elements of area tangent to the Grassmann image of an arbitrary submanifold of $E^N$, expressed in terms of the second quadratic forms of this submanifold.
The fundamental system of equations of an immersion of $L^n$ in $E^{2n-1}$ is investigated. Immersions of $L^3$ in $E^5$ under which one family of lines of curvature is composed of $L^3$ geodesics of are considered.
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