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What would this term correspond to? Inverse metric of connection.

  1. Nov 18, 2013 #1
    Suppose we are given two projection operators H' and H'' such that H' + H'' = 1, i.e. that any vector can be written as V = V' + V'' = (H' + H'') V.
    In a formula for a projection of the Riemann tensor (see the thread "Projection of the Riemann tensor formula") I encountered the term

    $$\left<X'\cdot B'(Y'',.), Z' \cdot B'(.,V'')\right>$$

    where ##B'(Y'',X'') = - \nabla'_{Y''}X''## and ##\left<\alpha, \beta\right> = g^{-1}(\alpha,\beta)## stands for the scalar product between the two one-forms ##\alpha = X'\cdot B'(Y'',.)## and ##\beta = Z' \cdot B'(.,V'')##, with ##g^{-1}## the inverse metric. I would like to translate this to something more recognizable (possibly something that looks like projection of terms corresponding to the Riemann tensor), but I'm stuck. I would guess that we would have that the term also can be written as

    $$\left<X'\cdot \nabla'_{Y''}, Z' \cdot \nabla' V''\right>$$

    but I do not see where to go from here. Any help will be appreciated!
  2. jcsd
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