Coframe
In mathematics, a coframe or coframe field on a smooth manifold [math]\displaystyle{ M }[/math] is a system of one-forms or covectors which form a basis of the cotangent bundle at every point.[1] In the exterior algebra of [math]\displaystyle{ M }[/math], one has a natural map from [math]\displaystyle{ v_k:\bigoplus^kT^*M\to\bigwedge^kT^*M }[/math], given by [math]\displaystyle{ v_k:(\rho_1,\ldots,\rho_k)\mapsto \rho_1\wedge\ldots\wedge\rho_k }[/math]. If [math]\displaystyle{ M }[/math] is [math]\displaystyle{ n }[/math] dimensional a coframe is given by a section [math]\displaystyle{ \sigma }[/math] of [math]\displaystyle{ \bigoplus^nT^*M }[/math] such that [math]\displaystyle{ v_n\circ\sigma\neq 0 }[/math]. The inverse image under [math]\displaystyle{ v_n }[/math] of the complement of the zero section of [math]\displaystyle{ \bigwedge^nT^*M }[/math] forms a [math]\displaystyle{ GL(n) }[/math] principal bundle over [math]\displaystyle{ M }[/math], 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. doi:10.3390/universe5100206. Bibcode: 2019Univ....5..206T.
See also
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