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I Riemannian Metric Invariance

  1. Apr 9, 2016 #1
    Q1: How do we prove that a Riemannian metric G (ex. on RxR) is invariant with respect to a change of coordinate, if all we have is G, and no coordinate transform?
    G = ( x2 -x1 )
    ( -x1 x2 )

    Q2: Since the distance ds has to be invariant, I understand that it has to be proved independantly of a specific coordinate transform. Any relationships between a given Riemannian metric and coordinate transforms?

    Thank you
     
    Last edited: Apr 9, 2016
  2. jcsd
  3. Apr 9, 2016 #2

    andrewkirk

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    The best approach to defining Riemannian metrics is coordinate-free, as a map that takes two vectors in the tangent space and returns a scalar. It then follows automatically that the metric is coordinate-invariant.

    For your specific case, you need to show that, for an arbitrary 2x2 change of coordinate matrix L and vectors ##u,v\in\mathbb{R}^2##:
    $$u^TGv=u'^TG'v'$$
    where the ##T## superscript denotes transpose and the 'primed' items are transformed to the new basis via ##L##, that is:

    $$u'=Lu,\ v'=Lv,\ G'=(L^{-1})^TGL^{-1}$$
     
  4. Apr 10, 2016 #3
    Very clear. Thank you very much Andrewkirk
     
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