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 $\mathbb{P}V$ is the geometry of 1, 2, and 3-dimensional subspace of V. The “points” of $\mathbb{P}V$ are the 1-dimensional subspaces, the “lines” are the 2-dimensional subspaces, and finally the “planes” are the 3-dimensional subspaces. We can describe a point of $\mathbb{P}V$ by using homogeneous coordinates, which is one of the most beautiful innovations ever introduced into analytic geometry (and was apparently introduced by Möbius). We write $(w:x:y:z)$ for a point of $\mathbb{P}(V)$, where not all of $w,x,y,z$ are zero. This makes sense since a representative vector for a one-dimensional subspace is a nonzero 4-tuple. Two points written in this notation are equal if we can multiply one by a nonzero scalar to obtain the other; hence why we have used colons within the notation in analogy with the ratio representation of a rational number. If we think of a plane of $\mathbb{P}V$ as the orthogonal complement of a point of $\mathbb{P}V$, then it makes sense to use the coordinates $[A:B:C:D]$ for a plane. This notation is further reinforced by noticing that the equation for the plane is $Aw+Bx+Cy+Dz=0$. Straight away we see one of the conveniences of homogeneous coordinates; a natural duality of $\mathbb{P}V$: $(w:x:y:z)\leftrightarrow [w:x:y:z]$.

Which leaves lines. Julius Plücker was Klein’s doctoral supervisor, and it was his observation that a line of $\mathbb{P}V$ could be represented by six scalars, that eventually gave rise to the Klein Correspondence. A line $L$ is spanned by two vectors $(u_0,u_1,u_2,u_3)$ and $(v_0,v_1,v_2,v_3)$. So let us look at the $2\times 4$ matrix where we place one of these vectors above the other:

$M_{u,v}:=\begin{bmatrix}u_0&u_1&u_2&u_3\\v_0&v_1&v_2&v_3\end{bmatrix}$

Now L is the row-space of $M_{u,v}$, and so this matrix has rank 2. So every $2\times 2$ minor of this matrix has nonzero determinant. So we take 4 choose 2 (=6) determinants from this matrix and list them. Define

$p_{ij}=\begin{vmatrix}u_i&u_j\\v_i&v_j\end{vmatrix}$

and take the 6-tuple of subdeterminants:

$(p_{01}:p_{02}:p_{03}:p_{12}:p_{31}:p_{23}).$

These are Plücker coordinates for $L$. What would happen if we had a different representation of $L$? If we instead took $(u_0',u_1',u_2',u_3')$ and $(v_0',v_1',v_2',v_3')$, then we can write the $4\times 2$ matrix from these two vectors as $M_{u,v}X$ for some invertible $2\times 2$-matrix $X$. And then the Plücker coordinates would be

$\det X\cdot (p_{01}:p_{02}:p_{03}:p_{12}:p_{31}:p_{23})$

by the “multiplicativite” property of the determinant. So we see that the Plücker coordinates of $L$ give a unique representation of $L$ if we regard them as homogenous coordinates of a point of a 5-dimensional projective space!

The matrix

$\begin{bmatrix}u_0&u_1&u_2&u_3\\v_0&v_1&v_2&v_3\\ u_0&u_1&u_2&u_3\\v_0&v_1&v_2&v_3\end{bmatrix}$

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

$p_{01}p_{23}+p_{02}p_{31}+p_{03}p_{12}=0$.

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

$Q(x_0,x_1,x_2,x_3,x_4,x_5):=x_0x_5+x_1x_4+x_2x_3$.

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 $\mathbb{P}V$ and points lying on this quadric. Moreover, this bijection, known as the Klein Correspondence, intertwines the concurrency relation of the lines with the orthogonality relation of the points of the Klein quadric. By orthogonality, I mean that if we take the bilinear form arising from the quadratic form, namely

$B(x,y):=Q(x+y)-Q(x)-Q(y)$

then two points represented by vectors $x$ and $y$ are orthogonal if $B(x,y)=0$. In other words, the line spanning $x$ and $y$ would have all of its points lying on the quadric, and so if we define such lines to be “lines of the quadric”, then we see that orthogonality is really “collinearity by lines of the quadric”. Likewise, there are planes of the five-dimensional projective space that are fully contained in the Klein quadric, such as

$\Pi: x_1=x_3=x_5=0$   and   $P:x_2=x_4=x_6=0$.

Just like the cross-product and more

The Plücker map has a striking similarity to the cross-product of vectors in $\mathbb{R}^3$, which is obtained by taking the 3 choose 2 (=3) subdeterminants of a $2\times 3$-matrix. The connection between these two notions lies in the exterior product of a vector space with itself. Formally, we define $V\wedge V$ to be the quotient vector space of $V\otimes V$ by the subspace of diagonal tensors $\{x\otimes x:x\in V\}$. In particular, this means that the induced product operation (by the tensor operation) is an alternating product called the exterior or wedge product. It has the (defining) property that $x\wedge x=0$ for all elements of the vector space. Note that $\dim(V\wedge V)={\dim V\choose 2}$.

Some consequences of the Klein Correspondence

Since the Klein Correspondence is incidence preserving, two concurrent lines of $\mathbb{P}V$ are mapped to two collinear points of the Klein quadric $\mathsf{Q}^+(5,q)$. If we take all the lines incident with a single point $p$ of $\mathbb{P}V$, then the image is a set of mutually collinear points of $\mathsf{Q}^+(5,q)$, that is, a plane of the quadric. Likewise, if we take all the lines contained in a single plane $\pi$ of $\mathbb{P}V$, then the image is again a plane (for the same reason). So points and planes of $\mathbb{P}V$ both yield planes of $\mathsf{Q}^+(5,q)$, but there is a catch. Two points of $\mathbb{P}V$ span a line, and so the corresponding planes must meet in a point. It is not difficult to see that an incident point and plane must be mapped to the set of points on a line of the quadric, and a non-incident point and plane yield two planes of the Klein quadric that are disjoint. In other words, we have an equivalence relation on planes of the Klein quadric that divides them into two equivalence classes:

Equivalence relation on planes of $\mathsf{Q}^+(5,q)$: 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 $\mathbb{P}V$ are mapped to the latins, whereas the planes are mapped to the greeks.

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, $\mathrm{P\Gamma L}(3,F):\langle \tau\rangle\cong \mathrm{P\Gamma O}^+(6,F)$, where $\tau$ is the standard duality given above.

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