In mathematics, a coframe or coframe field on a

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...

M

is a system of

one-form In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ...

s or

covector In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...

s which form a

basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting o ...

of the

cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This ...

at every point. In the

exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...

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

n

dimensional a coframe is given by a section

\sigma

\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 In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equ ...

over

M

, which is called the coframe bundle.

References

See also