“Projective Geometry Is All Geometry”

The title  is a quote attributed to  the mathematician William Cayley.

 

The goal of this post is to merely organize and make sense of the fact that all three of the principal continuous geometries are contained in projective geometry. Apriori, this is really remarkable since the definition of projective space (and its symmetries) as  linear subspaces of \mathbb R^n has very little to do with the other geometries we consider.  For me, the most interesting fact here is not that the underlying spaces (euclidian, spherical etc.) can be embedded in projective space, but rather that there exist embeddings that preserve the symmetries of each respective geometry, a statement that will be made precise immediately:

Definition 1: A pair (Y:H) is said to be a geometry (in the sense of Klein) if X is a set equipped with a transformation group G acting on X bijectively.

Definition 2: A geometry (X:G) is said to be a subgeometry of (Y:H) if there exists some inclusion i:X \hookrightarrow Y and a monomorphism \gamma : G \to H compatible with the group actions ( i.e i(x) \gamma(g)=i(xg) for x \in X and g \in G.)

Now, we can actually state our claims formally, although we will restrict our attention to \mathbb R P^2, and follow the methods used by Alexei Sossinsky in his short book “Geometries.” Some parts of the proof will be less detailed, but I’ll try to make clear the idea, and give explicit constructions of i and \gamma to prove the statements.

Claim 1: (\mathbb R^2: \mathrm{Isom}(\mathbb R^2)) is a subgeometry of (\mathbb R P^2: \mathrm{Proj}(2)), where \mathrm{Proj}(2) is the action of GL(3) modulo an equivalence relation that identifies all matrices whose column vectors differ by some constant muliple.

Alright, I will admit that this kind of claim is signifcantly less sexy than the quotation we are alleging to follow, so I will ammend statement 1 in the following way:

Claim 1.1: Euclidian geometry is easy (once you know everything there is to know about projective geometry.)

Proof: The idea here is extremely natural if one keeps in minds the homogeneous co-ordinate system. We view \mathbb RP^2 as

    \[\{x_1,x_2,1) \in \mathbb R^3\} \cup \Lambda_{\infty},\]

where \Lambda_\infty is the “line at infinity.”

With this in mind, we can define i: \mathbb R^2 \hookrightarrow \mathbb RP^2 so that (x_1,x_2) \mapsto (x_1:x_2:1).

As for the monomorphism, let O be the origin in \mathbb R^3, and choose some orthonormal frame in \mathbb R^2 given by two lines \lambda_1,\lambda_2. We let \gamma:G \to H be the monomorphism that sends the three lines OA,OB,OC to Og(A),Og(B),Og(C) for any g \in \mathrm{Isom}(\mathbb R^2), which proves the claim.

Claim 2: The hyperbolic plane , (H^2:M) is a subgeometry of the project plane.

There will be no claim 2.1, but instead we require a quick detour into the Cayley-Klein model for the hyperbolic plane. We let H^2 be the open disk in \mathbb R^2, and take consider the Lobachevsky metric,

    \[d(x,y)=|\mathrm{ln}(A,B,X,Y)|,\]

where X,Y are the intersection points of the line AB with S^1, the boundary of H^2 (commonly known as the absolute.) It turns out that the collection of projective transformations that map H^2 \to H^2, denoted P_0 is precisely the isometry group of H^2.

Since H^2 \subseteq \mathbb R^2, we can use the same inclusion as in claim 1. In order to define the monomorphism \lambda: \mathrm{Isom}(H^2) \to \mathrm{Proj}(2), we essentially just want to extend the action of P_0 on H^2 to th whole plane: any line XY \subseteq H^2 can be identified with the line in \mathbb R P^2 passing through inclusion of X and Y into \mathbb R P^2. Any parallel lines in H^2 correspond to lines that intersect at \Lambda_{\infty} in \mathbb R P ^2.

To be a bit more careful, we take 4 points in H^2 (in general position) and follow them under the action of g \in \mathrm{Isom} H^2. Since g is a projective transformation, these will suffice to define a homomorphism as follows:

We take the inclusion of all four points after g is applied to them, and define \lambda(g) to be the transformation i(X) \mapsto i(Xg). We know from the general theory of projective space, that there is exactly one such transformation for any 4 points, and thus we have a well-defined monomorphism (injective since such a transformation is unique and a homomorphism for clear reasons.) Showing that this is indeed a subgeoemetry amounts to proving that \gamma(g) agrees with g on the restriction to H^2, which follows since projective transformations preserve cross ratios for any 4 colinear points.

Claim 3: Riemannian elliptic geometry, (S^2/\sim,O(2)) where antipodal points are identified, is also a subgeometry of \mathbb R P^2.

Similarly to the previous 2 geometries, we want to think of \mathbb R P^2 as the usual euclidian plane along with the “line at infinity.” We will again embed the underlying space into the plane consisting of points of the form (x_1:x_2:1). However, unlike before, we need to be a little more clever. All we do is place the sphere above the plane, so that it is tangent at the southern hemisphere, and take pairs (X,-X) to the projection along the line passing through X and the center of the sphere. Note that this takes “lines” in elliptic geometry (which are greater circles) to lines, with the equator being mapped to the “line at infinity”” \Lambda_{\infty}. Spherical triangles will be mapped to triangles, but the angles will not be preserved. The monomorphism is constructed as before, by following the projection of 4 points and applying the general theory for projective transformations to take one transformation to the corresponding one in \mathbb RP^2.

The last proof was definitely sketchy, but it’s not terrible to show, and perhaps the constructions are more interesting than the proofs themselves since we can see “how” the geometry is embedded.