RP^2 vs CP: Comparing Stenographic Representations

[Icosahedron]

What is the relationship betwenn RP^2 and CP?

Espesicially, why are their stenographic representations different?

As far as I understand the stenographic representation for RP^2 goes like that:
a sphere with antipodal points identified is put above the R^2 plane, lines through the origin of the sphere cut the R^2 plane and project the sphere to the plane and so on.

Whereas, the stenographic representation for CP is: take a sphere, cut it in the middle with the R^2 plane and take lines starting from the north pole projecting from the sphere to the plane. And take lines starting from the south pole projecting from the sphere to the other side of the plane. Plus, transistion functions connecting these two mappings.

 

[Icosahedron]

Why has CP one additional point at infinity whereas RP^2 has infiinitely many points at infinity, one for each set of parallel lines?

Look here and here.

 

[Icosahedron]

Heeeellllloooooo!

Why is no one answering?

 

[Dragonfall]

Well, your links don't work, for one thing. Also I don't know what RP^2 and CP are, please elaborate.

 

[Icosahedron]

CP stands for complex projective line which is identical to the Riemann sphere.

RP^2 is the real projective plane which comes in many disguises. One way I described in post 1.

My questions:- why are their stenographic constructions so different?
- why has CP one point at infinity whereas RP^2 has one point at
infinity for each set of parallel lines
- what is the connection between CP and RP^2

 

[matt grime]

What is CP? It is

C^2\{(0,0)}/~

where ~ is the relation (u,v)~(tu,tv) where t is in C.
We can also identify it with the complex sphere, Cu{infinty}. As a Riemann surface it has two charts in the natural way of thinking about it. The two copies of C labelled V and U say have coordinates

[1;v] and [u;1]

Your 'point at infinity' is actually choosing a decomposition of CP as U u{(0,1)} with (0,1) the point at infinity.

Now, RP^2 has many descriptions, two two which are useful here are

R^3\{(0,0,0)}/~

where ~ is the relation (x,y,z)~(tx,ty,tz) where t is in R, it is a real manifold, with natural charts we'll call XY,XZ,YZ with coords

[x;y;1], [x;1;z] and [1;y;z] resp.

Your points at infinity now come from choosing XY as your affine open subset of interest and noting that the points in RP^2, and not in XY are of the form

[a;b;0]

where a,b are both not zero. We can identify [a;b;0] with the lines through the origin in R^2 - just map [a;b;0] to the line with slope a/b, where we understand that if b=0 we mean the a axis.

However RP^2 is also S^2 with antipodal points identified. The Riemann sphere is homeomorphic as a topological space, and indeed diffeomorphic as real manifolds (a complex n dim riemann surface is a real 2n dimensional manifold).

Thus there is a 2:1 mapping from RP^2 to S^2. In fact it is the universal cover, and the homotopy group of RP^2 is Z/2Z.

That enough? All I did was tell you what the definitions mean. Also, you can't link to files on your c drive of your desktop machine...

 

[Icosahedron]

Thanks Matt Grime. As you might have guessed, my mathematical maturity isn't that high, so I will read your post slow and carefully. But many thanks so far.

The links are
http://en.wikipedia.org/wiki/Projective_space

http://en.wikipedia.org/wiki/Riemann_sphere