Reading through an introductory Loop Quantum Gravity paper, I am given an induced Riemannian (space) metric:(adsbygoogle = window.adsbygoogle || []).push({});

\begin{equation}

q_{AB} = e^a_A e^b_B \delta_{ab}

\end{equation}

where $A = 1,2,3$ are covariant indices and $a = 1,2,3$ are internal indices of the triads $e^a_A$. The densitized triad is defined to be:

\begin{equation}

E^A_a := \frac{1}{2} \epsilon^{ABC}_{abc} e^b_B e^c_C.

\end{equation}

How do I determine the inverse metric $q^{AB}$ in terms of $E^A_a$? Must I guess the form and require that:

\begin{equation}

q^{AB}q_{BC} = \delta^A_C \ \ ?

\end{equation}

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# Relating inverse metric to densitized triads

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