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We can think of $\operatorname{SO}(3)$ as a bundle with base $S^2 = \operatorname{SO}(3)/\operatorname{SO}(2)$ and fibers $\operatorname{SO}(2)$. There is a well-defined projection $\pi : \operatorname{SO}(3) \rightarrow S^2$ that assigns to each $g \in \operatorname{SO}(3)$ its left coset, which can be thought of as a point on the sphere. In this context, a section is a map $\sigma : S^2 \rightarrow \operatorname{SO}(3)$ that has the property $\pi(\sigma(x)) = x$.

In general, global continuous sections may not exist, and I suspect they don't in this case. I suspect that this has something to do with the hairy ball theorem, which precludes global continuous non-zero vector fields, but that seems like something slightly different from a map $\sigma : S^2 \rightarrow \operatorname{SO}(3)$. So: do such global continuous sections exist, and if not why not? What is the relation to the hairy ball theorem?

If such global sections do not exist, are there any sections that are defined almost everywhere?

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