The expression of volume of asymptotic parallelepiped

Yu. A. Aminov

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 9 (2002), pp. 519-532.

Résumé

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 $\omega_i$.