Frame of a vector bundle ($e=(e_1,\dots ,e_r)$)

A frame over an open set $U$ is a list of smooth sections $e_1,\dots ,e_r$ of a rank-$r$ vector bundle whose values form a basis in every fibre over $U$. It is equivalent to a vector bundle trivialization by $(x,v)\mapsto {\sum }_a{e}_a(x)v^a$. Two local frames differ by a smooth map $g:U\to \mathrm{GL}(r)$, written $e^{\prime }=eg$.

For a vector bundle carrying a smooth nondegenerate symmetric bilinear form of constant metric signature, a pseudo-orthonormal frame is a smooth local frame of a vector bundle whose Gram matrix is the constant diagonal signature matrix $\eta$. It identifies the metric bundle locally with a fixed model ${\mathbb{R}}^{p,q}$. Two such frames differ by a smooth map into the corresponding orthogonal group. Positive-definite signature is included as the ordinary orthonormal case. A global pseudo-orthonormal frame is a section of a fiber bundle of the orthonormal frame bundle and trivializes that principal bundle.

A coframe dual to a local frame of the tangent bundle is the corresponding local frame of the cotangent bundle. Its elements are differential one-forms. Applying the coframe to covariant derivatives of the frame extracts the connection one-forms; applying it to the torsion tensor extracts the torsion forms.