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My first post.

  1. May 3, 2003 #1
    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.
    Last edited: May 5, 2003
  2. jcsd
  3. Jun 11, 2003 #2


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    Re: My first post, about rotation groups..

    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.
  4. Jun 11, 2003 #3
    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.
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