The fidelity between two (single-qubit) quantum states can be easily translated into the euclidean distance between the two states on the Bloch sphere (hilbert-schidmit distance). I'm curious if this concept can be extended to unitaries. If we have two single-qubit unitary matrices, denoted as $U_1$ and $U_2$, is there a way to interpret the fidelity (or any other metric) between these matrices with respect to the Bloch sphere?
For example, consider the case of $U_1 = X$ and $U_2 = I$. $U_1$ transforms $\lvert 0 \rangle$ into $\lvert 1 \rangle$ and $\lvert 1 \rangle$ into $\lvert 0 \rangle$ on the Bloch sphere, whereas $U_2$ leaves each basis state unchanged. Perhaps one approach to assessing the distance between these two unitary matrices is to sum the differences in fidelity between the states evolved by each matrix. In other words, $2 - F(U_1 \lvert 0 \rangle, U_2 \lvert 0 \rangle) + F(U_1 \lvert 1 \rangle, U_2 \lvert 1 \rangle)$, where $F$ represents fidelity. In this example, $F = 2$.
So far, it appears feasible to establish a relationship between the metric of two unitaries in terms of the Bloch sphere. However, if we set $U_1 = Z$, then $2 - F(U_1 \lvert 0 \rangle, U_2 \lvert 0 \rangle) + F(U_1 \lvert 1 \rangle, U_2 \lvert 1 \rangle) = 0$, since $Z$ has no effect on either $\lvert 0 \rangle$ or $\lvert 1 \rangle$.
Hence, my question is whether it's ever possible to think about the closeness (the metric can be arbitrary, including HS distance, unitary fideltiy, etc) of the given two unitaries, $U_1$ and $U_2$, in terms of the Bloch sphere?
(This question is inspired by this recent post: Sufficient conditions for a single-qubit unitary to be the identity)