Coframe

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