Hi,
I need to know if this relation is correct for a metric:
$g_{a[b}g_{c]d}=\frac{1}{2}\epsilon_{ace}\epsilon_{bdf}gg^{ef}$
I know that :
$\frac{1}{2}\epsilon_{ace}\epsilon_{bdf}g^{ef}=g_{b[a}g_{c]d}$
but I don't see how the determinant $g$ of the metric could appear.
Edit:
Ok so the previous relation emerged when computing the area of a surface $S$ in terms of the "densitized" triad $E_{i}^{a}=ee_{i}^{a}$ where $a,b,c,...$ are the spatial coordinates and $i,j,k,...$ are $SU(2)$ coordinates, e
the determinant of the triad matrix defined by $g_{ab}=e_{a}^{i}e_{b}^{j}\delta_{ij}$ where $g_{ab}$ is the spatiale metric. So, since the computation of the area uses the determinant of the the metric $h_{\alpha\beta}$ induced by $g_{ab}$ on $S$: ($\alpha,\beta,... =1,2\;and\; a,b,..=1,2,3$)
$h_{\alpha\beta}=g_{ab}\frac{\partial x^{a}}{\partial\sigma^{\alpha}}\frac{\partial x^{b}}{\partial\sigma^{\beta}}$
So in computing the determinant $h$ explecitely on finds the term
$g_{a[b}g_{c]d}$ which needs to equal to $\frac{1}{2}\epsilon_{ace}\epsilon_{bdf}gg^{ef}$ in order to obtain the final result:
$h=E_{i}^{a}E_{j}^{b}\delta^{ij}n_{a}n_{b}$ where $n$ are normal vectors $n_{a}=\epsilon_{abc}\frac{\partial x^{b}}{\partial\sigma^{1}}\frac{\partial x^{c}}{\partial\sigma^{2}}$
EDIT2:
After the notification of Willie Wong, I decided to put my original problem as a question, i.e: deriving the expression of the determinant of the induced metric on $S$ in terms of the densitized triad.