View source for Bianchi congruence

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''$B$-congruence''

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:

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

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]]].

====References====
<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>

[[Category:Geometry]]