Klein quadric

For the Klein quartic, see Klein quartic.

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 a line in S lie on a hyperbolic 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 underlying vector space the 6-dimensional exterior square Λ²V of V. The line coordinates obtained this way are known as Plücker coordinates.

These Plücker coordinates satisfy the quadratic relation

p₁₂p₃₄ + p₁₃p₄₂ + p₁₄p₂₃ = 0

defining Q, where

p_ij = u_iv_j − u_jv_i

are the coordinates of the line 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:

1. The points of S are the planes in C.
2. The lines of S are the points of Q.
3. 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 A₃ and D₃.

References

• Ward, Richard Samuel; Wells, Raymond O'Neil, Jr. (1991), Twistor Geometry and Field Theory, Cambridge University Press, ISBN 978-0521422680 .