I thought that my question would be very elementary and I could find the answer online somewhere, but I have been searching for it and still I have not found anything.
What is the relation between the hyperspherical and the Hopf coordinates for the unit three sphere?
Let's be more precise.
Starting from the four dimensional Euclidean flat space
\begin{equation}
ds^2 = \sum^3_{i=0} dx^2_i
\end{equation}
$\textbf{Hyperspherical coordinates}$
We can write a unit three-sphere simply by embedding it as
\begin{equation}
\begin{aligned}
x_0 &= \cos \chi_1 \, ,\\
x_1 &= \sin \chi_1 \cos \chi_2 \, ,\\
x_2 &= \sin \chi_1 \sin \chi_2 \cos \chi_3 \, ,\\
x_3 &= \sin \chi_1 \sin \chi_2 \sin \chi_3 \, ,\\
\end{aligned}
\end{equation}
and substituing the above yields
\begin{equation}
ds^2 = d \chi^2_1 + \sin^2 \chi_1 (d \chi^2_2 + \sin^2 \chi_2 d \chi^2_3)
\end{equation}
The coordinates take values as $0 \leq {\chi_1, \chi_2} \leq \pi$ and $0 \leq \chi_3 \leq 2 \pi$.
$\textbf{Hopf coordinates}$
These are the cooridnates for the embedding of $S^3$ in the complex plane $\mathbb{C}^2$. However, the coordinate change is easily implemented as an embedding in four dimensional Euclidean space as
\begin{equation}
\begin{aligned}
x_0 &= \cos \xi_1 \sin \vartheta \, , \\
x_1 &= \sin \xi_1 \sin \vartheta \, , \\
x_2 &= \cos \xi_2 \cos \vartheta \, , \\
x_3 &= \sin \xi_2 \cos \vartheta \, , \\
\end{aligned}
\end{equation}
and substituing the above once more in the invariant line element gives us
\begin{equation}
ds^2 = d \vartheta^2 + \sin^2 \vartheta d \xi^2_1 + \cos^2 \vartheta d \xi^2_2
\end{equation}
The above is the Hopf bundle
\begin{equation}
S^1 \rightarrow S^3 \rightarrow S^2
\end{equation}
The coordinates range as follows: $0 \leq \vartheta \leq \pi/2$ and $0 \leq {\xi_1, \xi_2} \leq 2 \pi$
How are these two coordinate sets related to one another?
Best Answer
Notice the last two coordinates imply $\chi_3=\xi_2$. Indeed, the hyperspherical coordinate system is just the usual 3D spherical coordinate system (with the $x_0$-axis the polar axis) but revolved using $x_2x_3$-plane rotations, and similarly the Hopf coordinate system is also a 3D spherical coordinate system (with $x_2$-axis the polar axis) also revolved using $x_2x_3$-plane rotations. Thus, it suffices to examine a 3D cross section:
$$ \begin{array}{rrrrr} x & = & \cos\chi_1 & = & \cos\xi\sin\vartheta \\ y & = & \sin\chi_1\cos\chi_2 & = & \sin\xi\sin\vartheta \\ z & = & \sin\chi_1\sin\chi_2 & = & \cos\vartheta \end{array} $$
Again, these are both just spherical coordinates, but with perpendicular polar axes. They become different coordinate systems in 4D since the first coordinate system is rotated between a fourth dimension and an axis perpendicular to the polar one (the $z$-axis), while the second coordnate system is rotated between a fourth dimension and the polar axis (again the $z$-axis).
Besides $\chi_3=\xi_2$, we can solve for one coordinates in terms of the other pretty easily by solving for parameters in the correct order. For instance, to get $\chi_1,\chi_2$ solve for $\chi_1$ first then $\chi_2$, and to get $\xi,\vartheta$ solve for $\vartheta$ first and then $\xi$ second.
If you stereographically project the two spherical coordinate systems (using the $x$-axis as the polar axis, say), you get polar coordinates for the first system and bipolar coordinates for the second.