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

• Manuel Tecchiolli (2019). "On the Mathematics of Coframe Formalism and Einstein-Cartan Theory -- A Brief Review". Universe. 5(10) (Torsion Gravity): 206. arXiv:2008.08314. Bibcode:2019Univ....5..206T. doi:10.3390/universe5100206.