Pfaffian of curvature tensor, reference request or explanation.

Question

For a Riemann manifold of dimension $2$, what is the Pfaffian of the curvature tensor?

Answer 1

OK, so when $M$ is a $2$-dimensional oriented Riemannian manifold, we have the skew-symmetric $2\times 2$ curvature matrix $\Omega_{ij} = \begin{bmatrix} 0 & \Omega_{12} \\ -\Omega_{12} & 0 \end{bmatrix}$, with $\Omega_{12} = \pm K\omega_1\wedge\omega_2$ (the sign depending on how you've set up your moving frames and the sign in the definition of the curvature 2-form). But the Pfaffian, by the definition you linked, is just $\frac 12(\Omega_{12} - \Omega_{21}) = \Omega_{12}$. Done.