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. In the exterior algebra of $M$, one has a natural map from $v_k:\stackrel{k}{⨁}{T}^{\ast }M\to \stackrel{k}{\bigwedge }{T}^{\ast }M$, given by $v_k:({\rho }_1,\dots ,{\rho }_k)\mapsto {\rho }_1\wedge \dots \wedge {\rho }_k$. If $M$ is $n$ dimensional, a coframe is given by a section $\sigma$ of $\stackrel{n}{⨁}{T}^{\ast }M$ such that $v_n\circ \sigma \ne 0$. The inverse image under $v_n$ of the complement of the zero section of $\stackrel{n}{\bigwedge }{T}^{\ast }M$ forms a $GL(n)$ principal bundle over $M$, which is called the coframe bundle.

References

• 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.

See also

• Frame fields in general relativity
• Moving frame

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