Coframe

In mathematics, a '''coframe''' or '''coframe field''' on a smooth manifold <math>M</math> is a system of one-forms or covectors which form a basis of the cotangent bundle at every point.<ref>{{Cite web |title=Structure coefficients of a coframe |url=https://math.stackexchange.com/questions/2722792/structure-coefficients-of-a-coframe |access-date=2024-01-19 |website=Mathematics Stack Exchange |language=en}}</ref> In the exterior algebra of <math>M</math>, one has a natural map from <math>v_k:\bigoplus^kT^*M\to\bigwedge^kT^*M</math>, given by <math>v_k:(\rho_1,\ldots,\rho_k)\mapsto \rho_1\wedge\ldots\wedge\rho_k</math>. If <math>M</math> is <math>n</math> dimensional, a coframe is given by a section <math>\sigma</math> of <math>\bigoplus^nT^*M</math> such that <math>v_n\circ\sigma\neq 0</math>. The inverse image under <math>v_n</math> of the complement of the zero section of <math>\bigwedge^nT^*M</math> forms a <math>GL(n)</math> principal bundle over <math>M</math>, which is called the coframe bundle.

==References== {{Refbegin}}

* {{cite journal | last1 = Manuel Tecchiolli | date = 2019 | title = On the Mathematics of Coframe Formalism and Einstein-Cartan Theory -- A Brief Review | journal = Universe | volume = 5(10) | issue = Torsion Gravity | pages = 206 | doi = 10.3390/universe5100206 | arxiv = 2008.08314 | bibcode = 2019Univ....5..206T | doi-access = free }} {{Refend}}

Category:Differential geometry