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Riemann tensor in normal coordinates

  1. Apr 2, 2010 #1
    This is essentially a "homework question", but I'm not looking for an explicit solution so I have posted it here.

    1. The problem statement, all variables and given/known data

    Find a simplified expression for the Riemann tensor in terms of the connection in normal coordinates.

    2. Relevant equations

    Riemann tensor = (derivative of connection term) - (derivative of connection term) - (connection term)(connection term) - (connection term)(connection term)

    3. The attempt at a solution

    My solution is

    Riemann tensor = (derivative of connection term) - (derivative of connection term)

    , where I have used the fact that the connections evaluated at point P are all 0, but their derivatives are not necessarily 0.

    My problem is that 3 MARKS are allocated to this question (from a possible 60 marks in a 2 hour paper), and that this looks far too simple a solution for 3 marks.

    What am I missing?

  2. jcsd
  3. Apr 2, 2010 #2
    A 3-point question of this type tells you that you have to first determine what the general form of the Riemann tensor is in normal coordinates and then expand the connections to obtain a four-term expression for the Riemann tensor where all terms are made of the second derivatives of the metric tensor.

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