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Difference between revisions of "Bianchi surface"

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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
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A surface of negative Gaussian curvature $K$ which can be expressed in asymptotic parameters $(u,v)$ 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>
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$$K=-\frac{1}{[U(u)+V(v)]^2},$$
  
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.
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where $U(u)$ and $V(v)$ are arbitrary functions; thus, Bianchi surfaces can be characterized by the fact that the function $(-K)^{-1/2}$ 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>
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$$\frac{\partial^2(-K)^{-1/2}}{\partial u\partial v}=0.$$
  
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" />).
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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 $U$ and $V$ are constant and the bent surface is a [[Voss surface|Voss surface]] (the class $B_0$).
  
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]].
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The class $B_1$ is characterized by the fact that only one family of lines of the principal base are geodesics (one of the functions $U,V$ is constant); conoids may serve as an example. The class $B_2$ corresponds to functions $U,V$ which both depend non-trivially on their arguments. See also [[Bianchi congruence|Bianchi congruence]].
  
  

Latest revision as of 14:19, 29 April 2014

A surface of negative Gaussian curvature $K$ which can be expressed in asymptotic parameters $(u,v)$ as

$$K=-\frac{1}{[U(u)+V(v)]^2},$$

where $U(u)$ and $V(v)$ are arbitrary functions; thus, Bianchi surfaces can be characterized by the fact that the function $(-K)^{-1/2}$ is diagonal with respect to its asymptotic net, i.e.

$$\frac{\partial^2(-K)^{-1/2}}{\partial u\partial v}=0.$$

For instance, a ruled Bianchi surface is a conoid — a surface attached to the 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 and to classify them. Thus, if the principal base contains two families of geodesic lines, the functions $U$ and $V$ are constant and the bent surface is a Voss surface (the class $B_0$).

The class $B_1$ is characterized by the fact that only one family of lines of the principal base are geodesics (one of the functions $U,V$ is constant); conoids may serve as an example. The class $B_2$ corresponds to functions $U,V$ which both depend non-trivially on their arguments. See also 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

[a1] L. Bianchi, "Lezioni di geometria differenziale" , 2 , Zanichelli , Bologna (1927) pp. Chapt. 1
[a2] S.P. Finikov, "Theorie der Kongruenzen" , Akademie Verlag (1959) (Translated from Russian)
How to Cite This Entry:
Bianchi surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bianchi_surface&oldid=11564
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article