This article is cited in
1 paper
The expression of volume of asymptotic parallelepiped
Yu. A. Aminovab a B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Khar'kov
b Institute of Mathematics, Bialystok University, 2 Academicka Str., 15-267, Bialystok, Poland
Abstract:
For an isometric immersion of a domain of Lobachevsky space
$L^n$ into Euclidean space
$E^{2n-1}$ there exists a coordinate net formed by asymptotic lines. Applying this net, we construct an
$n$-dimensional parallelepiped
$P$ called asymptotic. Properties of the volume
$V$ of
$P$ are considered in this paper. If
$n=2$, then there is the well-known Hazidakis formula for
$V$. By analogy with the case
$n=2$, J. D. Moore conjectured that the volume
$V$ could be calculated in terms of angles
$\omega_i$ between asymptotic curves at the vertices of
$P$ and that it is bounded from above. We obtain an expression of
$V$ for universal coverings of three- and four-dimensional analogues of pseudo-sphere and prove that
$V$ is bouded from above by an universal constant. On the other hand, we prove that there exist isometric immersions of domains of
$L^3$ into
$E^5$ so that it is impossible to express the volume
$V$ as an alternated sum of values of one function of two arguments dependent on angles
$\omega_i$.
MSC: 53A07 Received: 08.02.2002
Fulltext:
PDF file (261 kB)