Comparing C^2 to R^4: Complex Lines vs Real Planes

[bsaucer]

I'm trying to learn about the "complex plane" C^2, having two complex dimensions, which is supposedly like R^4, which has four real dimensions. I would assume there is a one-to-one correspondance between points in C^2 and the points in R^4.

My question at this point is about comparing the "complex lines", C^1, in C^2, and the "real planes", R^2, in R^4. Is there a one-to-one correspondance between the C^1's in C^2 and R^2's in R^4? Or are there more planes in R^4 than "complex lines" in C^2?

 

[Hurkyl]

Well, a complex line is the solution set to an equation such as

az + bw = c​

where (z,w) are the (complex) coordinates, and a,b,c are complex.

If we break z and w into real and imaginary parts, what is the corresponding equation form for those? Can an equation for every plane be put into that form?




However, I think you can do better if you're clever. Any two complex points determine a unique complex line, right? What does that statement translate to in R^4? Is it a true statement?

 

[bsaucer]

I was already thinking about the one line through two points part. Let me use terminology by Parker Manning. I would guess that C^2 would include include all planes in R^4, that include:

A reference plane
All planes parallel to the reference plane
And planes isocline to the reference plane in a certain sense (left or right, but not both).

In this set, any two planes are either parallel or isocline in the same sense, including planes that are absolutely perpendicular to each other. Any one of the planes in the set could be the "reference plane" mentioned above.

The set would not include:
Planes that intersect in a line
Planes that are "half-parallel" (or "skew")
Planes that are isocline in the "wrong" sense.

Am I right? Or am I on the wrong track?

 

[bsaucer]

One more question: Is it meaningful to talk about R^1's (real lines) and R^3's (real hyperplanes) as figures within C^2? How would those sets compare to the same figures in R^4? And is there such a thing as "half a complex dimension"?

 

[Hurkyl]

Isocline is not a term I'm familiar with, at least used this way. I haven't thought about how a real description of complex lines would look, beyond simply splitting complex linears equation into pairs of real equations.


As for subsets of real dimension 1 or 3, (AFAIK) those don't really play a part in complex algebraic geometry. I believe the same is true in the analytic analog too.

 

[bsaucer]

Two planes (with a point in common) are "isocline" if every ray in one plane (emanating from the common point) forms a constant angle with the other plane. This constant angle can be acute or right. When it's right, the planes are absolutely perpendicular.

If two isocline planes meet at the origin, they intersect the unit hypersphere about the origin in a pair of equidistant great circles (Clifford parallels).

Two planes in general are not usually isocline. Assuming they meet at a point, the acute angle between them can vary from a minimum value to a maximum value as the ray in one plane progresses around the point. If the maximum angle is right, the planes are half-perpendicular. A pair of non-isocline planes would intersect the unit hypersphere in two great circles that were not equidistant.