Families of submanifolds of constant negative curvature of many-dimensional Euclidean space

A family of $n$-dimensional submanifolds of constant negative curvature $K_0$ of the $(2n-1)$-dimensional Euclidean space $E^{2n-1}$ is considered and included in an orthogonal system of coordinates. For $n=2$ such a system of coordinates was considered by Bianchi. The concept of a many-dimensional Bianchi system of coordinates is introduced. The following result is central in the paper.
Theorem 1. {\it Assume that a ball of radius $\rho$ in the Euclidean space $E^{2n-1}$ carries a regular Bianchi system of coordinates such that $K_0\leqslant -1$. Then}
$$ \rho\leqslant\frac\pi4\,. $$

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