I am absolutely not a specialist in topology, and this question is quite hard for me... I read about metric tensors, geodesic formulas, but it is difficult for me to understand how to use them in order to solve this question.
Let $\mathbf{x}$ be a point on a manifold $S^2$ thas has coordinates $(\theta,\phi)$, with with $\theta$ being colatitude defined on $[0, \pi]$ and $\phi$ the azimuth defined on $[0, 2\pi]$. I know that for two points $\mathbf{x}_a, \mathbf{x}_b$ on a sphere, the distance between them is
$$d=\arccos \left( \cos \theta_a \cos \theta_b + \sin \theta_a \sin \theta_b \cos \left(\phi_a - \phi_b\right) \right)$$
I am now looking for the same problem, but for the distance $d$ between two points $\mathbf{x}_a, \mathbf{x}_b$ on a manifold $S^2 \times S^2$.