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General Expression for Round Metric on an N-sphere

  1. Jun 4, 2014 #1
    1. The problem statement, all variables and given/known data
    I want to know the expression for the round metric of an n-sphere of radius r


    2. Relevant equations

    I have obtained this for a 3-sphere
    dS^{2}=dr^{2}+r^{2}(d\theta_{1}^{2}+sin^{2}\theta_{1}d\theta_{2}^{2}
    +sin^{2}\theta_{1}sin^{2}\theta_{2}d\theta_{3}^{2})


    3. The attempt at a solution
    what will be the general expression for an n -sphere
     
  2. jcsd
  3. Jun 5, 2014 #2
    Looks good to me except that [itex]dr^2[/itex] shouldn't be part of it since the radial direction is not a direction on the n-sphere :smile:

    Examining the results for n=1, 2 and 3 should reveal the pattern for general n...
     
  4. Jun 7, 2014 #3
    What is a round metric(in words)? And an induced metric(in words)?
    (I dont know the tag for hyperlinks)

    http://en.wikipedia.org/wiki/Metric_tensor#The_round_metric_on_a_sphere

    http://en.wikipedia.org/wiki/Induced_metric

    The benefit of using spherical coordonates(on ##\Re^n##)

    http://en.wikipedia.org/wiki/N-sphere#Spherical_coordinates

    is that the induced metric(on "##S^n##") is just the restriction of the metric for r=cst.
    How did you get your answer?
    For an arbitrary dimension you can use the coordinate transformations rule for tensors(you know the metric of ##\Re^n## in cartesian coordinate, you know the transformations between them and the spherical coordinates)
     
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