# Euler density of two-dimensional manifolds

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

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

$$E_2=2R_{1212}$$

(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

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

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

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

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

I) It seems the resolution to OP's question lies in the difference between

1. the Levi-Civita symbol, which is not a tensor and whose values are only $0$ and $\pm 1$; and

2. the Levi-Civita tensor, whose definition differs from the Levi-Civita symbol by a factor of $\sqrt{|\det(g_{\mu\nu})|}$.

II) The 2D Euler-density is

$$E_2~=~ \frac{1}{8\pi} \epsilon_{\mu\nu}\epsilon_{\lambda\kappa} R^{\mu\nu\lambda\kappa}~=~ \frac{1}{8\pi} \epsilon^{\mu\nu}\epsilon^{\lambda\kappa} R_{\mu\nu\lambda\kappa}~=~\frac{S}{4\pi}~=~\frac{K}{2\pi},$$

where $S=2K$ is the scalar curvature and $K$ is the Gaussian curvature. The Euler characteristics is

$$\chi~=~\int d^2x \sqrt {| \det g_{\mu\nu} |} E_2.$$

• Is there a reference from which I can get this information, and perhaps other related information as well? Commented Sep 16, 2015 at 12:51