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I am reading a proof of Pu's inequality in Katz' book "Systolic geometry and topology". In section 6.4, he describes a fibration $q : SO(3) \to S^2$ which I can't quite figure out.

First, he thinks of $SO(3)$ as the unit tangent bundle of $S^2$ (the round $S^2$ in $\mathbb{R}^3$) by identifying $g \in SO(3)$ with $dg(v) \in TS^2$ for some fixed $v \in TS^2$ of unit length. He then says that a fiber of $q$ corresponds to an orbit of the geodesic flow on $SO(3)$. So a fiber of $q$ over a point is the collection of unit tangent vectors over some (directed) great circle in $S^2$.

He says that for this fibration, we should think of the base space as the configuration space of all directed great circles in $S^2$. So I guess that if we could see $S^2$ as that configuration space, we could just send $g \in SO(3)$ (viewed as a unit tangent vector) to the directed great circle to which it is pointing. However, I don't see how to identify $S^2$ with its directed great circles, anyone has an idea?

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A great circle is the same data as two antipodal points, namely the two points furthest away from the circle. That is, the space of great circles is the space of two antipodal points on $S^2$, which is $\mathbb{RP}^2$. The direction picks one of these points (say, the point that appears on the left as you follow the direction around the circle), which corresponds to the double cover $S^2 \to \mathbb{RP}^2$.

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If you have a directed circle, use the right-hand rule that would position your thumb in the appropriate direction. That direction will give the right point on the sphere.

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