Pullbacks that annihilate the Maurer-Cartan form

Question

Let $\mathcal{G}$ be the group that consists of $C^{\infty}\left(M,G\right)$ equipped with the binary operation of pointwise multiplication, where $M$ is a ($\infty$-connected) smooth manifold (in my case $\mathbf{R}^4$) and $G$ a compact Lie group. I am interested about the set of elements $\sigma\in\mathcal{G}$ for which $\sigma^*\mu_G=0$, where $\mu_G$ is the Maurer-Cartan form on $G$. The obvious subset of $\mathcal{G}$ that satisfies this condition is that of constant maps, which is isomorphic to $G$. Besides the elements of that set, are there any other elements in $\mathcal{G}$ whose pullback annihilates $\mu_G$?

Answer 1

Since $\mu_G$ is a pointwise isomorphism, $\sigma^* \mu_G = 0$ implies that the differential of $\sigma$ is identically zero. Since you're assuming $M$ is connected, that implies $\sigma$ is constant.