Bianchi congruence

$B$-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)