### Geometric Help Wanted

#### Posted by David Corfield

I’m giving a talk to some philosophers and would like to get across in a non-technical way the idea that geometry has gone, and continues to go, through profound changes. I wanted to touch on the Erlangen Program, and perhaps illustrate its idea with the following example:

Elementary plane geometry and the projective investigation of a quadric surface with reference to one of its points are one and the same.

Klein’s explanation is as follows:

If a quadric surface be brought into correspondence with a plane by stereographic projection, the surface will have one fundamental point, - the centre of projection. In the plane there are two, - the projections of the generators passing through the centre of projection. It then follows directly: the linear transformations of the plane which leave the two fundamental points unaltered are converted by the representation (

Abbildung) into linear transformations of the quadric itself, but only into those which leave the centre of projection unaltered. By linear transformations of the surface into itself are here meant the changes undergone by the surface when linear space-transformations are performed which transform the surface into itself. According to this, the projective investigation of a plane with reference to two of its points is identical with the projective investigation of a quadric surface with reference to one of its points. Now, if imaginary elements are also taken into account, the former is nothing else but the investigation of the plane from the point of view of elementary geometry. For the principal group of plane transformations comprises precisely those linear transformations which leave two points (the circular points at infinity) unchanged.

David Rowe explains this as follows:

Consider the stereographic projection $f: S \to \mathbb{P}^2 (\mathbb{C})$ from a fixed point $P$ on a 2nd-degree surface $S$ in $\mathbb{P}^3 (\mathbb{C})$. This mapping is one-to-one except for two points $p$ and $q$ in the range which are the image of two generators of $S$ that pass through $P$. Now the group of linear transformations of $\mathbb{P}^2 (\mathbb{C})$ that leave $p$ and $q$ fixed, when pulled back by $f^{-1}$, yields the group of linear transformations of $S$ that fix $P$, i.e., the restrictions to $S$ of those linear transformations of $\mathbb{P}^3 (\mathbb{C})$ that leave both $S$ and $P$ invariant. Furthermore, since any two points of $\mathbb{P}^2 (\mathbb{C})$ are projectively equivalent, one can take $p$ and $q$ to be the circular points at infinity. But, by Cayley’s principle, the linear transformations of $\mathbb{P}^2 (\mathbb{C})$ that leave these points fixed are precisely the transformations of Euclidean plane geometry. (The Early Geometric Works of Sophus Lie and Felix Klein, 261)

He goes on to complain that the “vast majority of writers” derive their knowledge of the Program from secondary sources, which

…trivialize its content, in particular by restricting their discussion to examples in which familiar transformation groups act on real spaces. This interpretation necessarily overlooks all the deeper geometrical results–which are the only ones Klein bothered to present–since these all require that the base space be a complex manifold. (p. 264)

So what’s happening here? There’s an isomorphism between the group of Euclidean transformations and the subgroup of $PGL(3, \mathbb{C})$ which fixes the quadric and a point on it? Any more illumination appreciated.

## Re: Geometric Help Wanted

The automorphism group of the quadric is SO(4) having Dynkin diagram D_2 which are just 2 points, that is the same as A_1xA_1. Geometrically this says that the quadric is isomorphic to P^1 x P^1 (the two families of lines on the quadric). If you fix a point on the quadric you also fix the two lines on it through that point, so the complement is A^1 xA^1 = A^2 (here the A^k is k-dml affine space) the affine plane.