Connection one-form ($A$)

Write a local frame of a rank-$r$ vector bundle as the row $e=(e_1,\dots ,e_r)$. The connection one-form of a connection on a vector bundle is the matrix of one-forms $A$ determined byThe placement of $A$ matters: its entry $A^i{}_j$ is the coefficient of $e_i$ in $\nabla e_j$. If $e^{\prime }=eg$, substitution in the Leibniz rule givesThe derivative term means that $A$ is not itself a global endomorphism-valued one-form. The curvature form $F=dA+A\wedge A$ instead changes by $F^{\prime }=g^{-1}Fg$, and the difference of two connections on a vector bundle has the same homogeneous change law. These formulas fix the row-frame convention consistently.