Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

When do SO(2) actions on the circle in the plane determine a metric?

  1. Mar 5, 2013 #1


    User Avatar
    Science Advisor
    Gold Member
    2017 Award

    a metric on the plane determines an action of SO(2) on is unit circle by rotation.

    Suppose one starts with a free transitive action of SO(2) on a circle. When does this come from a metric? Always?
  2. jcsd
  3. Mar 5, 2013 #2
    Here is a rough idea without calculations. I have not proved anything, so I might be misunderstanding something somewhere, or I might have just made a mistake in my thinking. Also, since you did not state the nature of the metric, I assumed the easiest thing which is that the metric is an inner product metric. Please clarify if you meant something more general.

    Lets first consider a free transitive action of SO(2) on the circle. Both can be denoted R/2piZ.

    Then we can parametrize these actions by strictly increasing functions F(s) satisfying F(0)=0, F(2pi)=2pi and whose derivative F' is periodic. Then the action of s on t is given by

    [itex] s*t = F(s+F^{-1}(t)) [/itex]

    Now let us consider the action induced by a metric. This is vague because I dont' know what you mean exactly by metric. So I will assume you mean an inner product metric. This is given by a quadratic form: Ax^2+2Bxy+Cy^2. I think it is clear that the resulting action of SO(2) is unchanged by scaling the coefficients of this inner product, so such actions are determined by 2 numbers. In more geometric terms, these two numbers are:
    1. The angular inclination of the eigenvector that lies in the first quadrant,
    2. The ratio of the eigenvalues.

    If your metric is Finsler, then I don't understand it well enough to describe the resulting action of SO(2) on the circle. But I would be interested to know more about that.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: When do SO(2) actions on the circle in the plane determine a metric?
  1. Determinant 2 (Replies: 5)