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Conformal mappings and special networks of Weyl spaces. (English) Zbl 1032.53004
Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 4th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6-15, 2002. Sofia: Coral Press Scientific Publishing. 239-247 (2003).

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 $\left(n>2\right)$. If any net $\left({v}_{1},{v}_{2},\cdots ,{v}_{n}\right)$ 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 $\left(n>2\right)$. If any net $\left({v}_{1},{v}_{2},\cdots ,{v}_{n}\right)$ 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 $\left(n>2\right)$. If any net $\left({v}_{1},{v}_{2},\cdots ,{v}_{n}\right)$ 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.

Reviewer: A.Neagu (Iaşi)
##### MSC:
 53B15 Other connections 53C40 Global submanifolds (differential geometry)