A Metric of the "space" of 3d rotations

[pervect]

I was recently reading that the space of 3d rotations should have the topology of a real projective space. For confirmation, see wiki, https://en.wikipedia.org/wiki/3D_rotation_group.

The Lie group SO(3) is diffeomorphic to the real projective space P 3 ( R ) . {\displaystyle \mathbb {P} ^{3}(\mathbb {R} ).}[4]

Consider the solid ball in R^3 of radius π (that is, all points of R^3 of distance π or less from the origin). Given the above, for every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point from the origin. The identity rotation corresponds to the point at the center of the ball. Rotation through an angle 𝜃 between 0 and π (not including either) are on the same axis at the same distance. Rotation through angles between 0 and −π correspond to the point on the same axis and distance from the origin but on the opposite side of the origin. The one remaining issue is that the two rotations through π and through −π are the same. So we identify (or "glue together") antipodal points on the surface of the ball. After this identification, we arrive at a topological space homeomorphic to the rotation group.

It seems to me that when we assign coordinates to this space (I was thinking of using the Euler angles, but actually there's no need to be so specific), the resulting space should have a metric, the "distance" between points being the amount one has to rotate to get from one "point" to another.

I was wondering if anyone has written a metric for this space (there should be more than one, I'm interested in any such realization), and whether it would be Riemannian or pseudo-Riemanian. There is also the possibility that my intuition that such a metric exists is incorrect, proof that it does not exist would also be interesting.

 

[PeterDonis]

when we assign coordinates to this space (I was thinking of using the Euler angles, but actually there's no need to be so specific), the resulting space should have a metric

I don't think this is true in general; assigning coordinates means it's a manifold, but a manifold does not have to have a metric.

 

[renormalize]

I was wondering if anyone has written a metric for this space (there should be more than one, I'm interested in any such realization), and whether it would be Riemannian or pseudo-Riemanian. There is also the possibility that my intuition that such a metric exists is incorrect, proof that it does not exist would also be interesting.

This paper might be germane: Metrics for 3D Rotations: Comparison and Analysis.

 

[Orodruin]

There is the concept of the Haar measure, but that relates to a group invariant volume form, not necessarily to a metric tensor. I am unsure whether similar arguments can be used to construct a metric tensor (i.e., a positive definite symmetric bilinear map on the Lie algebra so(3)). I have not seen such a construction, but that does not mean it doesn't exist.

Edit: Search and thou shalt find https://en.wikipedia.org/wiki/Killing_form

 

[robphy]

Possibly useful: (I haven’t read any of them)

https://en.wikipedia.org/wiki/Charts_on_SO(3)

https://rotations.berkeley.edu/geodesics-of-the-rotation-group-so3/