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

1. Nov 18, 2013

### center o bass

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!