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Difference between revisions of "Bianchi congruence"

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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016010/b0160102.png" />-congruence''
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''$B$-congruence''
  
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A congruence of straight lines in which the curvatures of the focal surfaces at the points situated on the same straight line of the congruence are equal and negative. The principal surfaces of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016010/b0160103.png" />-congruence cut out conjugate line systems on its focal surfaces. The straight lines of the congruence map the asymptotic nets of the focal surfaces onto an orthogonal net on a sphere. The curvature of a focal surface of a Bianchi congruence is expressed in asymptotic parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016010/b0160104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016010/b0160105.png" /> by the formula:
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A [[Congruence of lines|congruence of straight lines]] in which the curvatures of the focal surfaces at the points situated on the same straight line of the congruence are equal and negative. The principal surfaces of a $B$-congruence cut out conjugate line systems on its focal surfaces. The straight lines of the congruence map the asymptotic nets of the focal surfaces onto an orthogonal net on a sphere. The curvature of a focal surface of a Bianchi congruence is expressed in asymptotic parameters $u$ and $v$ by the formula:
  
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<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/b016010/b0160106.png" /></td> </tr></table>
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$$K=\frac{1}{(\phi(u)+\psi(v))^2}.$$
  
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Surfaces whose curvatures satisfy this condition are called Bianchi surfaces (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016010/b0160108.png" />-surfaces, cf. [[Bianchi surface|Bianchi surface]]).
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Surfaces whose curvatures satisfy this condition are called Bianchi surfaces ($B$-surfaces, cf. [[Bianchi surface|Bianchi surface]]).
  
 
Bianchi congruences were studied by L. Bianchi [[#References|[1]]].
 
Bianchi congruences were studied by L. Bianchi [[#References|[1]]].
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Bianchi,  ''Ann. Mat. Pura Appl.'' , '''18'''  (1890)  pp. 301–358</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.P. Finikov,  "Theorie der Kongruenzen" , Akademie Verlag  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.P. Finikov,  "Bending and related geometrical problems" , Moscow-Leningrad  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.I. Zhulikovskii,  "Classical differential geometry in a tensor setting" , Moscow  (1963)  (In Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  L. Bianchi,  ''Ann. Mat. Pura Appl.'' , '''18'''  (1890)  pp. 301–358</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  S.P. Finikov,  "Theorie der Kongruenzen" , Akademie Verlag  (1959)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  S.P. Finikov,  "Bending and related geometrical problems" , Moscow-Leningrad  (1937)  (In Russian)</TD></TR>
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<TR><TD valign="top">[4]</TD> <TD valign="top">  V.I. Zhulikovskii,  "Classical differential geometry in a tensor setting" , Moscow  (1963)  (In Russian)</TD></TR>
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</table>
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[[Category:Geometry]]

Latest revision as of 16:33, 19 October 2014

$B$-congruence

A congruence of straight lines in which the curvatures of the focal surfaces at the points situated on the same straight line of the congruence are equal and negative. The principal surfaces of a $B$-congruence cut out conjugate line systems on its focal surfaces. The straight lines of the congruence map the asymptotic nets of the focal surfaces onto an orthogonal net on a sphere. The curvature of a focal surface of a Bianchi congruence is expressed in asymptotic parameters $u$ and $v$ by the formula:

$$K=\frac{1}{(\phi(u)+\psi(v))^2}.$$

Surfaces whose curvatures satisfy this condition are called Bianchi surfaces ($B$-surfaces, cf. Bianchi surface).

Bianchi congruences were studied by L. Bianchi [1].

References

[1] L. Bianchi, Ann. Mat. Pura Appl. , 18 (1890) pp. 301–358
[2] S.P. Finikov, "Theorie der Kongruenzen" , Akademie Verlag (1959) (Translated from Russian)
[3] S.P. Finikov, "Bending and related geometrical problems" , Moscow-Leningrad (1937) (In Russian)
[4] V.I. Zhulikovskii, "Classical differential geometry in a tensor setting" , Moscow (1963) (In Russian)
How to Cite This Entry:
Bianchi congruence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bianchi_congruence&oldid=19286
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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