One of the central and important concepts in projective geometry is the beautiful connection between 3-dimensional projective space and the Klein quadric. As is indicated by the title, this correspondence between these two geometries was named after the German mathematician Felix Klein, who studied it in his dissertation *Über die Transformation der allgemeinen Gleichung des zweiten Grades zwischen Linien-Koordinaten auf eine kanonische Form* (1868). The Klein Correspondence can be used to give geometric understanding for certain isomorphisms of low rank classical groups, and we will give an example of some of these in a follow-up post. In geometry, the Klein Correspondence can sometimes illuminate an obscure object in projective 3-space, for the Klein quadric is naturally embedded in a 5-dimensional projective space where there is an added richness to the geometry, and one has available other ways to distinguish certain configurations. For example, if you were to learn about linear complexes in 3-dimensional projective space, you might find one class known as the “parabolic congruences” as a somewhat messy object of pencils of lines centered on the points of a common line. However, under the Klein Correspondence, a parabolic congruence becomes a *quadratic cone*: one of the first 3-dimensional geometric objects we first encountered in school mathematics. Notice that I have not specified the field we are working over; it won’t matter!

### Coordinates for 3D projective space

Given a 4-dimensional vector space V over a field F, the projective space

Which leaves lines. Julius Plücker was Klein’s doctoral supervisor, and it was his observation that a line of

Now L is the row-space of

and take the 6-tuple of subdeterminants:

These are Plücker coordinates for

by the “multiplicativite” property of the determinant. So we see that the Plücker coordinates of

### A quadratic form

The matrix

has rank 2, and so has zero determinant. It then follows that

Therefore, the Plücker coordinates of a line yield a point of five-dimensional projective space that is a zero of the following quadratic form:

The zeros of this quadratic form define a non-singular* quadric * called the *Klein quadric*. Amazingly, the converse is true! The map defined by taking the Plücker coordinates yields a bijection between the lines of

then two points represented by vectors

### Just like the cross-product and more

The Plücker map has a striking similarity to the cross-product of vectors in **exterior product** of a vector space with itself. Formally, we define *exterior* or *wedge* product. It has the (defining) property that

### Some consequences of the Klein Correspondence

Since the Klein Correspondence is incidence preserving, two concurrent lines of

Equivalence relation on planes of: Equal or meet in a point.

The two equivalence classes are known as the *latins* and the *greeks*. So without loss of generality, the points of

One of the main consequences of the Klein Correspondence is that the correlation group of the projective 3-space (collineations together with dualities) and the collineation group of the Klein quadric, are isomorphic. That is,

This post is getting quite long now, so I’ll write some more on the consequences of the Klein Correspondence in a later post.