Coframe

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

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