I was reading this page of nlab: https://ncatlab.org/nlab/show/Maurer-Cartan+form
I do here a small recall first of the Maure-Cartan form first: If you have a Lie group $G$, then you have the Maurer Cartan form which sends each tangent vector to the tangent space of $G$ at the identity. It is given by the following construction: For any $a$ and $b$ in $G$, there exists exactly one $\theta(a,b)$ such that $a\cdot \theta(a,b)=b$. It implies that for a smooth curve $b_t$ through $a$, $\theta(a,b_t)$ is a smooth curve through the identity. So the tangent vector defined by $b_t$ is sent to the tangent vector defined by $\theta(a,b_t)$. This is how the Maurer-Cartan form is defined on a Lie group.
But in this page of nlab, they say in the introduction that the concept of the Maurer-Cartan form can be generalized to a $G$-principal bundle. Since $G$ acts freely and transitively on the fibres, it follows that we can do the same definition than before with the $\theta$, and so we can send any vertical tangent vector to a tangent vector of $G$ at the identity.
My question is: how is this Maurer-Cartan form defined for non-vertical tangent vectors? The article of nlab does not give further informations, so I do not know how to extend the definition. I'm also asking myself if there is only one Maurer-Cartan form on principal bundle.
Thank you for your help!
Best Answer
You need to fix a connection in your principal bundle. That is, you introduce what an horizontal vector is, and then you can project any vector field to a vertical one. In this way, you can project and then go to the Lie algebra of the group.