Consider the Euler class of a manifold $e(\mathcal{M})$ in 4D given by
$$e(\mathcal{M}) = \frac{1}{4\pi^2}\epsilon_{abcd}~\Omega^{ab}\wedge\Omega^{cd}$$ where $\Omega$ is the curvature two-form. I want to find the space of all connection such that $e(\mathcal{M}) = 0$. Has this been done before and if so how? And do these connections have any special names?