Riemann sphere in infinite dimensional space?

[Log]

Riemann "sphere" in infinite dimensional space?

I was just reading about the Riemann sphere, in 3-space and find it very interesting. With it you assign a point on the sphere to every location in the xy-plane. Then I thought, in 4-space you would be able to assign every point in 3-space to a point on the 4-dimensional Riemann "sphere".

Is there any such thing in infinite dimensional space? That would be pretty cool!

 

[dimension10]

I was just reading about the Riemann sphere, in 3-space and find it very interesting. With it you assign a point on the sphere to every location in the xy-plane. Then I thought, in 4-space you would be able to assign every point in 3-space to a point on the 4-dimensional Riemann "sphere".

Is there any such thing in infinite dimensional space? That would be pretty cool!

Riemannian geometry + Hilbert Geometry? Is that what you are asking? Then, maybe you want to know about Fock spaces. I don't know much of them myself, so just go to this wikipedia link: http://en.wikipedia.org/wiki/Fock_space.

Also, the Riemann Sphere can also be understood as the riemannian counterpart of the complex plane. http://en.wikipedia.org/wiki/Riemann_sphere#As_the_complex_projective_line

I know, that's a lot of wikipedia links, but wikipedia is correct unless someone's vandalised it, in which case, its quite obvious.