Coframe

In mathematics, a coframe or coframe field on a smooth manifold $M$ is a system of one-forms or covectors which form a basis of the cotangent bundle at every point.^[1] In the exterior algebra of $M$ , one has a natural map from $v_{k}:\bigoplus ^{k}T^{*}M\to \bigwedge ^{k}T^{*}M$ , given by $v_{k}:(\rho _{1},\ldots ,\rho _{k})\mapsto \rho _{1}\wedge \ldots \wedge \rho _{k}$ . If $M$ is $n$ dimensional, a coframe is given by a section $\sigma$ of $\bigoplus ^{n}T^{*}M$ such that $v_{n}\circ \sigma \neq 0$ . The inverse image under $v_{n}$ of the complement of the zero section of $\bigwedge ^{n}T^{*}M$ forms a $GL(n)$ principal bundle over $M$ , which is called the coframe bundle.