What is the equivalent of a circle in $\mathbb{C}^2$?

Question

One natural way to generalize the circle from the real plane $\mathbb{R}^2$ to the complex equivalent $\mathbb{C}^2$ is to use the same defining equation:

$$z^2+w^2=1$$

Where in this case the "radius" is 1 and the "center" is $(0,0)$. Of course if you think of $\mathbb{C}^2$ as $\mathbb{R}^4$, then geometrically this set looks nothing like a circle. It is unbounded and has no clear radius or center. It does have a lot of the same properties as the circle however. Any line intersects it at most twice, but unlike the real circle almost every line does intersect it twice. This is the kind of thing we tend to see when we take a real concept and extend it to the complexes: the same properties apply except nicer.

On the other hand we could argue that the natural extension of the circle is this set:

$$|z|^2+|w|^2=1$$

Geometrically viewing $\mathbb{C}^2$ as $\mathbb{R}^4$ again this set has the shape of the hypersphere $S^3$, which does feel more circle like. However it is uncirclelike in some other ways. For example, the first "circle" I gave can be parameterized as $(\sin(\theta),\cos(\theta)),\theta\in\mathbb{C}$ much like the real circle. On the other hand the second "circle" cannot be parameterized in a complex variable, and in fact is not even a complex manifold since it has 3 real dimensions.

My question is what is the name of each of these sets, and which one is the most natural extension of a circle to the complexes?

Answer 1

One can refer to $\{(x,y): x^2+y^2=1\}$ is "a nonsingular quadric in the affine plane". (All such quadrics over complex numbers are affine-isomorphic. You can also define it as $\{(z,w): zw=1\}$ for instance.)

You can safely refer to $\{(x,y)\in {\mathbb C}^2: |x|^2+|y|^2=1\}$ as the "unit sphere in ${\mathbb C}^2$".

As for which one is a natural generalization of the unit circle, it is all in the eye of a beholder. I would never refer to either one as a "complex circle".

Answer 2

Like I said in my comment, this is highly dependent on what you're trying to “extend.” As a differential geometer, I vote for $S^3$.

One interesting property of the circle is that it is a Lie group; that is, a manifold with a smooth group action. Algebraically, it is the group of all complex numbers with modulus $1$.

There is a four-dimensional real division algebra called the quaternions ($\mathbb{H}$), which looks like the complex numbers but has two additional imaginary units $j$ and $k$ (which do not commute; the rule is $ij = k = -ji$). There is a modulus on quaternions, too, and the group of unit quaternions is diffeomorphic to $S^3$ as a manifold. It's also algebraically isomorphic to the group $SU(2)$ of $2\times 2$ complex matrices $A$ such that $AA^*=I$ and $\det(A) = 1$.

Answer 3

I would say x^2 + y^2 = 1 is the set of all complex circles of radius unity while the the second parmeterization, namely modulus (x^2) + modulus (y^2) = 1, is a set of two complex functions whose sum of moduli is unity. I am again sorry here, as I am an economist, not a mathematician. Sincerely, Richard Anthony Baum