*(English)*Zbl 1032.53004

The authors prove the following theorems:

Theorem 1. If ${W}_{n}$ is a totally umbilical hypersurface of a recurrent Weyl space ${W}_{n+1}$, then ${W}_{n}$ is also conformally recurrent.

Theorem 2. Let a totally umbilical hypersurface ${W}_{n}$ of a recurrent Weyl space ${W}_{n+1}$ be conharmonically Ricci-recurrent $(n>2)$. If any net $({v}_{1},{v}_{2},\cdots ,{v}_{n})$ in ${W}_{n}$ is a Chebyshev net of first kind with respect to ${W}_{n+1}$, it is also a Chebyshev net of the first kind with respect to ${W}_{n}$ and the converse is also true.

Theorem 3. Let a totally umbilical hypersurface ${W}_{n}$ of a recurrent Weyl space ${W}_{n+1}$ be conharmonically Ricci-recurrent $(n>2)$. If any net $({v}_{1},{v}_{2},\cdots ,{v}_{n})$ in ${W}_{n}$ is a Chebyshev net of the second kind with respect to ${W}_{n+1}$, it is also a Chebyshev net of the second kind with respect to ${W}_{n}$ and the converse is also true.

Theorem 4. Let a totally umbilical hypersurface ${W}_{n}$ of a recurrent Weyl space ${W}_{n+1}$ be conharmonically Ricci-recurrent $(n>2)$. If any net $({v}_{1},{v}_{2},\cdots ,{v}_{n})$ in ${W}_{n}$ is a geodesic net with respect to ${W}_{n+1}$ it is also a geodesic net with respect to ${W}_{n}$ and conversely.