Gauss-Bonnet theorem in the Hawking/Ellis book

Question

At the page 336 of Hawking, Ellis: The Large Scale Structure of Space-Time, the Gauss-Bonnet theorem is stated as

$$\int_H \hat{R}\ d\hat{S} = 2\pi \chi(H) \qquad (1)$$

with

$$\hat{R} = R_{abcd} \hat{h}^{ac} \hat{h}^{bd}$$

and induced metric on the horizon $\hat{h}_{ab}$,

$$\hat{h}_{ab} = g_{ab} + \ell_a n_b + n_a \ell_b \ ,$$

where $\ell^a$ and $n^a$ is a pair of future-directed null vectors on the horizon.

Is there a missing factor of 2 on the RHS of equation (1)?

The (2-dimensional) Gauss-Bonnet theorem in the literature is usually stated using "Gaussian curvature" $K = R/2$, so I'm suspecting in this "hidden factor" (compare it, for example, with Heusler: Black Hole Uniqueness Theorems, equations (6.23)--(6.26)).

Answer 1

Yes, the factor $2\pi$ in eqn 1 should be $4\pi$ – if the area integral is normalized conventionally. For example, for a sphere, the scalar curvature is $R=2/a^2$. When multiplied over the $4\pi a^2$ surface, we get $8\pi$ which is $4\pi$ times the Euler character of the sphere, $\chi=2$. Well, at least I hope that these considerations aren't affected by using the metric $\hat h$ instead of $g$.