Connection one form in a local frame

Let $e=(e_1,\dots ,e_r)$ be a local frame of $E$ on an open set $U$, in the sense of Local and global frames of a vector bundle. Restrict a given connection to $U$ by A bundle connection is local and restricts to open sets. Its connection one-form in the frame $e$ is the matrix $\omega =({\omega }^i{}_j)$ of scalar one-forms on $U$ characterized by ${\mathrm{\nabla }}_X{e}_j={\sum }_i{\omega }^i{}_j(X)e_i\mathrm{.}$ For each $j$, the smooth Hom section $\mathrm{\nabla }{e}_j$ has unique fibre coordinates in $e$; these coordinates are linear in the direction and smooth, hence are scalar one-forms. Smoothness may be tested in any coordinate chart and frame, using Smoothness of a section is equivalent to smooth local components.

Our convention treats $e$ as a row and the coefficients $u$ of $s=eu$ as a column. A frame change is $e^{\mathrm{\prime }}=eA$ with a smooth invertible matrix $A$, so $u=A{u}^{\mathrm{\prime }}$ for the same section. Products of matrices and one-forms use ordinary matrix multiplication with scalar functions as coefficients. In a coordinate chart, ${\omega }^i{}_j={\sum }_k{\mathrm{\Gamma }}^i{}_{kj}d{x}^k$; this notation does not assert tensorial transformation for $\mathrm{\Gamma }$.

For rank zero the frame and connection matrix are empty; for rank one $\omega$ is a scalar one-form. On a zero-dimensional base every one-form is zero. The frame is supplied locally, so this definition involves no global selection or existence theorem.