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 $$v_k:\bigoplus^kT^*M\to\bigwedge^kT^*M$$, 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 $$\bigoplus^nT^*M$$ such that $$v_n\circ\sigma\neq 0$$. The inverse image under $$v_n$$ of the complement of the zero section of $$\bigwedge^nT^*M$$ forms a $$GL(n)$$ principal bundle over $$M$$, which is called the coframe bundle.