mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a coframe or coframe field on a smooth manifold

M

is a system of one-forms 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 ...

s which form a basis 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 may ...

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

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 over

M

, which is called the coframe bundle.

References

See also