Demoulin surface

A surface formed by a conjugate net of lines the tangents to which form two $W$-congruences — a so-called singular conjugate system. Only the Demoulin surfaces permit a projective deformation. Introduced by A. Demoulin .

References

[1a]	A. Demoulin, "Sur les surfaces $R$ et les surfaces $\Omega$" C.R. Acad. Sci. Paris , 153 (1911) pp. 590–593
[1b]	A. Demoulin, "Sur les surfaces $R$ et les surfaces $\Omega$" C.R. Acad. Sci. Paris , 153 (1911) pp. 705–707
[1c]	A. Demoulin, "Sur les surfaces $R$" C.R. Acad. Sci. Paris , 153 (1911) pp. 797–799
[1d]	A. Demoulin, "Sur les surfaces $\Omega$" C.R. Acad. Sci. Paris , 153 (1911) pp. 927–929
[2]	S.P. Finikov, "Projective-differential geometry" , Moscow-Leningrad (1937) (In Russian)
[3]	A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad (1976) (In Russian)

Comments

The terminology concerned with Demoulin surfaces differs. In [a1] they are roughly characterized by the fact that the Demoulin tetrahedron (see Demoulin quadrilateral) degenerates to one point. The existence of a projective deformation is a more general condition (see [a1], Par. 119) The problem of projective deformation is related to $R$-congruences, which are special $W$-congruences (see [a1] and [2], [3]).

References

[a1]	G. Bol, "Projective Differentialgeometrie" , 2 , Vandenhoeck & Ruprecht (1954)
[a2]	E.P. Lane, "A treatise on projective differential geometry" , Univ. Chicago Press (1942)