View source for Bianchi surface

A surface of negative Gaussian curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016030/b0160301.png" /> which can be expressed in asymptotic parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016030/b0160302.png" /> as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016030/b0160303.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016030/b0160304.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016030/b0160305.png" /> are arbitrary functions; thus, Bianchi surfaces can be characterized by the fact that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016030/b0160306.png" /> is diagonal with respect to its asymptotic net, i.e.

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016030/b0160307.png" /></td> </tr></table>

For instance, a ruled Bianchi surface is a [[Conoid|conoid]] — a surface attached to the [[Peterson surface|Peterson surface]]. If a Bianchi surface is given, it is possible to determine the classes of surfaces which can be obtained by a [[Deformation over a principal base|deformation over a principal base]] and to classify them. Thus, if the principal base contains two families of geodesic lines, the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016030/b0160308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016030/b0160309.png" /> are constant and the bent surface is a [[Voss surface|Voss surface]] (the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016030/b01603010.png" />).

The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016030/b01603011.png" /> is characterized by the fact that only one family of lines of the principal base are geodesics (one of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016030/b01603012.png" /> is constant); conoids may serve as an example. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016030/b01603013.png" /> corresponds to functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016030/b01603014.png" /> which both depend non-trivially on their arguments. See also [[Bianchi congruence|Bianchi congruence]].



====Comments====
The notion of a  "deformation over a principal base"  is not very common in Western literature, and this type of deformation has no standard name even. It can best be characterized as a deformation preserving a conjugate net.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Bianchi,   "Lezioni di geometria differenziale" , '''2''' , Zanichelli , Bologna  (1927)  pp. Chapt. 1</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.P. Finikov,   "Theorie der Kongruenzen" , Akademie Verlag  (1959)  (Translated from Russian)</TD></TR></table>