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:

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

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.