Euler density of two-dimensional manifolds

Question

I am asking this question after reading this post: What is Euler Density?.

For a two dimensional manifold, the Euler density is given by:

\begin{equation} E_2=2R_{1212} \end{equation}

(note that $R_{1212}$ is the only independent component of the Riemann tensor in 2d).

Now, integrating over the 2d manifold, we should get the Euler characteristic

\begin{equation} \chi=\int d^2x \sqrt {(\textrm{det }g)} E_2, \end{equation}

where $(\textrm{det }g)$ is determinant of the metric. But $E_2=2R_{1212}=R(g_{11}g_{22}-g_{12}g_{21})=R \textrm{ det }g$, where $R$ is the Ricci scalar of the 2d manifold. This gives

\begin{equation} \chi=\int d^2x (\textrm{det }g)^{\frac{3}{2}} R, \end{equation}

which contradicts the first term of equation 3.2.3b of Polchinski's 'String Theory', volume 1. What's the reason for this contradiction?

This post imported from StackExchange Physics at 2015-09-15 19:02 (UTC), posted by SE-user Meer Ashwinkumar