The definition of this 1-form $\gamma$?

Question

Let $M$ be a manifold on which acts a Lie group $G$.

Let $\sigma \in \mathcal{A}^1(M) \otimes \mathfrak{g}$ be a connection form for the principal bundle $M \rightarrow M/G$. We define a $G$- invariant 1-form $\gamma$ on $M × \mathfrak{g}^*$ in the following way $$(m,\xi) \in M × \mathfrak{g}^*, \gamma_{|(m,\xi)}:= \langle \sigma_{|m}, \xi\rangle.$$

I don't understand the definition of the 1-form $\gamma$: suppose we choose a tangent vector $X \in T_{(m,\xi)}(M × \mathfrak{g}^*)$, then what is $\gamma_{|(m,\xi)}(X)$ ?

Answer 1

In words, $\gamma$ is a $1$-form on $M\times \mathfrak{g}^*$ that works like this: $\gamma$ takes $(m,\xi)$ to a linear functional $\gamma_{(m,\xi)}: T_mM\times \mathfrak{g}^* \to \Bbb R$, and $\gamma_{(m,\xi)}$ takes $(v,\eta)\in T_mM\times \mathfrak{g}^*$ to the value of $\xi\in \mathfrak{g}^*$ evaluated at $\sigma_m(v)$. But the quantity $\xi(\sigma_m(v))$ is being denoted as $\langle \sigma_m(v),\xi\rangle$ (in context, $\langle \xi, \sigma_m(v)\rangle$ would also be acceptable).

This $\gamma$ is essentially a tautological form built from $\sigma$.