Unveiling the Connection: Rotation Groups & Hyperspheres

[Tyger]

My first post, about rotation groups..

A result about rotation groups.

To me this seems clear, simple and very intuitive, but in all the papers and books I've read on the subject I have never seen it presented. Maybe some of you have seen it, or maybe it is new. It is very simple to state:

The group of orthonormal rotations in a space of n dimensions, SO(n) is isomorphic to the group of geodesic translations in a positively curved space (hypersphere) of n(n-1)/2 dimensions.

 

[marcus]

Originally posted by Tyger
A result about rotation groups.

To me this seems clear, simple and very intuitive, but in all the papers and books I've read on the subject I have never seen it presented. Maybe some of you have seen it, or maybe it is new. It is very simple to state:

The group of orthonormal rotations in a space of n dimensions, SO(n) is isomorphic to the group of geodesic translations in a positively curved space (hypersphere) of n(n-1)/2 dimensions.

this is a nice thought. It may require a special clarification of what is meant by "group of geodesic translations" in order to make sense-----or this could be my private confusion and it is immediately understandable to everyone but me!

I think of the case n=3 where your theorem says
SO(3) is isomorphic to the geodesic translations of a sphere in 3 dimensions.
This seems right except that rotation around an axis is only a "geodesic translation" for points on the equator. So that one may have to extend the definition in some fashion.

Sorry about the vagueness, I just this moment saw your message and am replying directly.

 

[Tyger]

Maybe I better clarify my statement

Choose any two points in a sphere of n(n-1)/2 dimensions, draw a geodesic from one point to the other. Every such geodesic can be mapped to a rotation in a space of n dimensions.