What are some common metrics on an n-sphere $S = \{x \in \mathbb{R}^n : ||x||_2 = 1 \} $ ?
I think that the angle between vectors is a metric on $S$. What other common metrics are there? Citations also welcome.
Background: I have an estimation algorithm which theoretically works for any metric. However, in practice some metrics may have additional properties which I may be able to make use of algorithmically (e.g. for faster computation). However, the vector-angle-metric is the only one I know of.
There are surprisingly few sensible choices to consider. Symmetry considerations motivate any metric $d(x,\,y)$ reduce to one on the centre-$O$ circle passing through $x,\,y$, and further to some function of the angle $\theta\in[0,\,\pi]$ between them, which lies in that circle's plane. The most obvious options are the arc length $\theta$ (which you cited) and the chord length $\sqrt{2(1-\cos\theta)}=2\sin\tfrac{\theta}{2}$.