Mean Distance on a 3-sphere?

Question

What is the (analytical) mean geodesic distance between a set of randomly chosen points on a 3-sphere?

Answer 1

The volume element of a $(n-1)$-sphere is given by

$$d_{S^{n-1}}V = \sin^{n-2}(\phi_1)\sin^{n-3}(\phi_3)\cdots\sin(\phi_{n-2}) d\phi_1d\phi_2\cdots d\phi_{n-1}$$ and the admissible range for $\phi_i$ is $[0,\pi]$ for $1 \le i \le n-2$ and $[0,2\pi]$ for $i = n-1$.

So for a 3-sphere, the average distance is just

$$\frac{\int_0^{\pi} \phi \sin^2(\phi) d\phi}{\int_0^{\pi} \sin^2(\phi) d\phi} \tag{*}$$

Notice

$$\sin^2(\phi) = \sin^2(\pi - \phi) \quad\implies\quad \int_0^{\pi} \phi \sin^2(\phi) d\phi = \int_0^{\pi} (\pi - \phi) \sin^2(\phi) d\phi$$

It is then clear the average in $(*)$ is just $\frac{\pi}{2}$.

Alternatively, one can pick a point $p$ on $S^n$ as origin. The reflection of $S^n$ in the direction of $p$ in the ambient space $\mathbb{R}^{n+1}$:

$$S^n \ni \vec{x}\quad\longrightarrow\quad 2 (\vec{p}\cdot\vec{x}) \vec{p} - \vec{x} \in S^n$$

is an isometry of $S^n$ and sends a point at a distance $r$ from $p$ to a distance $\pi - r$ from $p$. Averaging over this sort of pairs always give us $\frac{\pi}{2}$.