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 $v_k:{⨁}^k{T}^{*}M\to {\bigwedge }^k{T}^{*}M$, 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 ${⨁}^n{T}^{*}M$ such that $v_n\circ \sigma \ne 0$. 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