// Workers AI · dad joke modeWhat did the coframe say? I'm framed.

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. 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.

See also

• Frame fields in general relativity
• Moving frame

References

Sources

• 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.