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 $(n>2)$. If any net $(v_1,v_2,\dots, v_n)$ 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 $(v_1,v_2,\dots, v_n)$ 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 $(v_1,v_2,\dots, v_n)$ 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. For the entire collection see [

Zbl 1008.00022].

##### MSC:

53B15 | Other connections |

53C40 | Global submanifolds (differential geometry) |