Let ${U,V}$ be two points on the orthogonal group $O(n)=\{X\in\mathbb{R}^{n\times n}: X^TX=I\}$ equipped with the usual Riemannian metric.
Let $d(U,V)$ be the geodesic distance between $U,V$ on $O(n)$. Do we have any analytic expression of $d(U,V)$? Let $\|\|_F$ be the Frobenius norm on $\mathbb{R}^{n\times n}$. Then we of course have $\|U-V\|_F\leq d(U,V)$. There must also exist a constant $C>0$, such that $d(U,V)\leq C\|U-V\|_F$. What is the best constant $C$ we can have?