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Coframe

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated July 1, 2026

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 ^{k}T^{*}M\to \bigwedge ^{k}T^{*}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 ^{n}T^{*}M such that v_{n}\circ \sigma \neq 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.

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This article is adapted from the Wikipedia article “Coframe”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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