Klein quadric

In mathematics, the lines of a 3-dimensional projective space, ''S'', can be viewed as points of a 5-dimensional projective space, ''T''. In that 5-space, the points that represent each line in ''S'' lie on a quadric, ''Q'' known as the Klein quadric.

If the underlying vector space of ''S'' is the 4-dimensional vector space ''V'', then ''T'' has as the underlying vector space the 6-dimensional exterior square Λ²''V'' of ''V''. The line coordinates obtained this way are known as Plücker coordinates.

These Plücker coordinates satisfy the quadratic relation
: $p_ p_+p_p_+p_ p_ = 0$
defining ''Q'', where
: $p_ = u_i v_j - u_j v_i$
are the coordinates of the line Linear span, spanned by the two vectors ''u'' and ''v''.

The 3-space, ''S'', can be reconstructed again from the quadric, ''Q'': the planes contained in ''Q'' fall into two equivalence classes, where planes in the same class meet in a point, and planes in different classes meet in a line or in the empty set. Let these classes be $C$ and $C'$. The geometry of ''S'' is retrieved as follows:

# The points of ''S'' are the planes in ''C''.
# The lines of ''S'' are the points of ''Q''.
# The planes of ''S'' are the planes in ''C''’.

The fact that the geometries of ''S'' and ''Q'' are isomorphic can be explained by the isomorphism of the Dynkin diagrams ''A''₃ and ''D''₃.

References

* Albrecht Beutelspacher & Ute Rosenbaum (1998) ''Projective Geometry : from foundations to applications'', page 169, Cambridge University Press
* Arthur Cayley (1873) "On the superlines of a quadric surface in five-dimensional space", ''Collected Mathematical Papers'' 9: 79–83.
* Felix Klein (1870) "Zur Theorie der Liniencomplexe des ersten und zweiten Grades", Mathematische Annalen 2: 198
* Oswald Veblen & John Wesley Young (1910) ''Projective Geometry'', volume 1, Interpretation of line coordinates as point coordinates in S₅, page 331, Ginn and Company.
* {{citation, title=Twistor Geometry and Field Theory, first1=Richard Samuel, last1= Ward, first2= Raymond O'Neil, Jr., last2= Wells, author2-link=Raymond O. Wells, Jr., publisher=Cambridge University Press, year= 1991, bibcode=1991tgft.book.....W, isbn=978-0-521-42268-0.

Projective geometry
Quadrics