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].

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].

Reviewer: A.Neagu (Iaşi)