I'm currently confused about the intrinsic metric of spheres. Is it equivalent to the angular distance, the geodesic distance, or the great-circle distance? Are these metrics all defined as $$ d(x, y)=\arccos{(x^{\top}y)} $$ for $x, y$ on a unit sphere $\mathcal{S}^{p-1}\subset\mathbb{R}^{p}$?
And for arbitrary $x, y\in \mathbb{R}^{p}$, can these metrics be represented as $$ d(x, y)=\arccos{\left(\frac{x^{\top}y}{\|x\|\|y\|}\right)}? $$