Gauss-Bonnet term in Physics

Question

Given a 4-dimensional compact manifold (torsion free), the Euler characteristic is defined as:

$$E_4 ~=~ \int \epsilon_{abcd}R^{ab} \wedge R^{cd}$$

with $R^{ab}$ is the curvature 2-form. Perturb the connection 1-form (represent by $\delta \omega^{ab}$), $E_4$ should be unchanged. How can I proof that, and what will be change if the manifold is not torsion free anymore?

Given that:

Connection 1-form from torsion-free condition:

$$T^a ~=~ De^a ~=~ de^a + \omega^a_b \wedge e^b ~=~ 0$$

and

$$\omega^{ab} ~=~ \delta^{ac} \omega^b_c.$$

Curvature 2-form:

$$R^{ab} ~=~ D\omega^{ab} ~=~ d\omega^{ab} + \omega^a_c \wedge \omega^{cb}.$$

Gauss-Bonnet term appears as a topological term in some theoretical gravitational actions. I don't see why it's unchanged, so I post this question of mine here.

Answer 1

I think you can try a variation with respect to $\omega_{ab}$. First of all $R^{ab} \neq D\omega^{ab}$ because $\omega^{ab}$ is not a tensor, the correct statement actually is $$\delta R^{ab} = D\delta\omega^{ab}$$ Using this, we may do a variation of $E_4$ with respect to $\omega^{ab}$, to obtain

$$\delta E_4 = -2\int \epsilon_{abcd} D R^{ab}\wedge \delta\omega^{cd}$$ I have used integration by parts to shift the covariant exterior derivative on $R^{ab}$. However, already $DR^{ab} = 0$ by Bianchi identity (https://en.wikipedia.org/wiki/Curvature_form). So, this shows that of $E_4$ is independent of any variation in $\omega^{ab}$. Notice I have not used any assumption about torsion here. My proof is independent of that.