View source for Darboux surfaces

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''wreath of''

Surfaces associated with an [[Infinitesimal deformation|infinitesimal deformation]] of one of them; discovered by G. Darboux [[#References|[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|Peterson correspondence]],  $  \mathbf z _ {i+} 1 $
and  $  \mathbf x _ {i-} 1 $
are in [[Polar correspondence|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Darboux,   "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''4''' , Gauthier-Villars  (1896)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Shulikovskii,   "Classical differential geometry in a tensor setting" , Moscow  (1963)  (In Russian)</TD></TR></table>

====Comments====
For the notion of a  $  W $-
congruence cf. [[Congruence of lines|Congruence of lines]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Fubini,   E. Čech,   "Introduction á la géométrie projective différentielle des surfaces" , Gauthier-Villars  (1931)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Bol,   "Projective Differentialgeometrie" , Vandenhoeck &amp; Ruprecht  (1954)</TD></TR></table>