Darboux surfaces

wreath of

Surfaces associated with an infinitesimal deformation of one of them; discovered by G. Darboux [1]. Darboux surfaces form a "wreath" of 12 surfaces, with radius vectors $\mathbf x _ {1} \dots \mathbf x _ {6} , \mathbf z _ {1} \dots \mathbf z _ {6}$ satisfying the equations

$$d \mathbf z _ {i} = [ \mathbf z _ {i + 1 } , d \mathbf x _ {i} ] ,\ \ d \mathbf x _ {i} = [ \mathbf x _ {i - 1 } , d \mathbf z _ {i} ] ,$$

$$\mathbf z _ {i} - \mathbf x _ {i + 1 } = [ \mathbf z _ {i+ 1 } , \mathbf x _ {i} ],\ i = 1 \dots 6 ,$$

$$\mathbf x _ {i + 6 } = \mathbf x _ {i} ,\ \mathbf z _ {i + 6 } = \mathbf z _ {i} ;$$

where $\mathbf z _ {i+} 1$ and $\mathbf x _ {i}$ are in Peterson correspondence, $\mathbf z _ {i+} 1$ and $\mathbf x _ {i-} 1$ are in polar correspondence, while $\mathbf z _ {i}$ and $\mathbf x _ {i+} 1$ are poles of a $W$- congruence. A similar "wreath" is formed by pairs of isometric surfaces of an elliptic space.

References

[1]	G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 4 , Gauthier-Villars (1896)
[2]	V.I. Shulikovskii, "Classical differential geometry in a tensor setting" , Moscow (1963) (In Russian)

Comments

For the notion of a $W$- congruence cf. Congruence of lines.

References

[a1]	G. Fubini, E. Čech, "Introduction á la géométrie projective différentielle des surfaces" , Gauthier-Villars (1931)
[a2]	G. Bol, "Projective Differentialgeometrie" , Vandenhoeck & Ruprecht (1954)