I'm attending a differential geometry course, and I'm stuck at one part of a question that we've been asked. The following rotated circle is given in 4 dimensions:
$$x_1x_3 + x_2x_4 = \frac{1}{2}$$
I need its 4-dimensional parameterization, using a $\phi \in [0,2\pi[$ value.
To show that this IS in fact a circle, you can solve the following system of equations to obtain their intersection:
$$ I. x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1$$ $$ II. v = z \iff x_1 + ix_2 = x_3 + ix_4$$
Where I. is a 4-dimensional sphere, II. is a plane in 4 dimensions, with the correspondance of $$\mathbb{C}^2 \simeq \mathbb{R}^4$$ $$z := x_1 + ix_2$$ $$v := x_3 + ix_4$$ I'm including the calculation of this intersetion, just to give you a deeper understanding of the problem, however my question only extends to how to obtain the parametric equation of the above circle.
Note: The intersection of a 4-dimensional sphere and a plane can only give you a 2-dimensional circle, since by definition a 4D sphere is the collection of points equal distance from the origin. When we intersect the sphere with a plane, we apply this definition again, so we must get a circle.
Note 2: As far as I understand, this is a circle that's been rotated 45°'s in the $x_3$, and 45°'s in the $x_4$ direction.
Now I'll include the above mentioned calculation:
$$II. \Rightarrow x_3 = x_1 + ix_2 - ix_4 = x_1 + i(x_2 - x_4)$$ $$I. \Rightarrow x_1^2 + x_2^2 + (x_1 + ix_2 - ix_4)^2 + x_4^2 = 1$$ $$\Rightarrow x_1^2 + x_2^2 + x_1^2 - x_2^2 - x_4^2 + 2ix_1x_2 - 2ix_1x_4 + 2x_2x_4 + x_4^2 = 1$$ $$2x_1^2 + 2(ix_1x_2 - ix_1x_4 + x_2x_4) = 1$$ $$x_1^2 + x_1i(x_2-x_4) + x_2x_4 = \frac{1}{2}$$ Since in the first equation of this calculation we assumed that $$x_3 = x_1 + i(x_2 - x_4) \iff x_3 - x_1 = i(x_2 - x_4)$$ We can substitute to get rid of the imaginary unit: $$x_1^2 + x_1(x_3 - x_1) + x_2x_4 = \frac{1}{2}$$ $$x_1^2 + x_1x_3 - x_1^2 + x_2x_4 = \frac{1}{2}$$ $$x_1x_3 + x_2x_4 = \frac{1}{2}$$ And we get our original equation.
To get back to my original question, I've tried the parameterization of: $$C(\phi) = \biggl(\frac{1}{\sqrt{2}}cos(\phi),\frac{1}{\sqrt{2}}cos(\phi),sin(\phi),sin(\phi)\biggr), \phi \in [0,2\pi[$$ But after a long while of calculation it ended up being wrong, because I've got nonsense. So dear Stack Exchange users, can you help me figure this one out?
The single equation you wrote at the beginning does not describe a circle, but a $3$-dimensional object. However, your own proof shows that the following is a circle:
$$ x_1x_3+x_2x_4=\frac12,\ x_1=x_3,\ x_2=x_4. $$
Or to write it even more simply
$$ x_1^2+x_2^2=\frac12,\ x_1=x_3,\ x_2=x_4. $$
In other words, you forgot the equality constraints. This should now not be a problem to parametrize.