View source for Projective normal

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A generalization of the concept of a [[Normal|normal]] in metric geometry. Distinct from the latter, where a normal is totally determined by the tangent plane to a surface (i.e. the first-order neighbourhood), this does not hold in projective geometry. Even terms of the first order of smallness do not determine the vertices of a coordinate tetrahedron not lying in the tangent plane (i.e. for a chosen [[Darboux quadric|Darboux quadric]] it is impossible to construct a single self-polar tetrahedron). This is a natural situation: the projective group is much larger than the group of motions, and therefore its invariants must be of higher order; but even in the fourth-order neighbourhood there is no unique straight line that could be taken as the third axis of a tetrahedron. In this way one obtains, e.g.:

the Wilczynski directrix

$$W=N+\frac1{\mathrm I}u^ir_i;$$

the Green edge

$$G=N+\frac14\left(g^{pq}\frac{\partial_q\mathrm I}{\mathrm I}-\frac1{\mathrm I}A_{ij}T^{ijp}\right)r_p;$$

the Čech axis

$$C=N+\frac13\left(g_{is}\frac{\partial_s\mathrm I}{2\mathrm I}-\frac1{\mathrm I}A_{jk}T^{ijk}\right)r_i;$$

and the Fubini normal

$$F=N+g^{is}\frac{\partial_s\mathrm I}{2\mathrm I}r_i.$$

Here $N$ is the [[Affine normal|affine normal]].

They all lie in one plane.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.A. Shirokov,   A.P. Shirokov,   "Differentialgeometrie" , Teubner  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.P. Norden,   "Spaces with an affine connection" , Nauka , Moscow-Leningrad  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.P. Finikov,   "Projective differential geometry" , Moscow-Leningrad  (1937)  (In Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Bol,   "Projektive differentialgeometrie" , '''I, II''' , Vandenhoeck &amp; Ruprecht  (1950)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Cartan,   "Leçons sur la théorie des espaces à connexion projective" , Gauthier-Villars  (1937)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Fubini,   E. Čech,   "Introduction á la géométrie projective différentielle des surfaces" , Gauthier-Villars  (1931)</TD></TR></table>