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Isometric immersions of domains of $n$-dimensional Lobachevsky space in $(2n-1)$-dimensional Euclidean space
Yu. A. Aminov
Abstract:
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.
Bibliography: 15 titles.
UDC:
513.82
MSC: Primary
53C42; Secondary
53A35 Received: 01.08.1979
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