One-form (differential geometry)
Differential form of degree one or section of a cotangent bundle / From Wikipedia, the free encyclopedia
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In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle.[1] Equivalently, a one-form on a manifold is a smooth mapping of the total space of the tangent bundle of to whose restriction to each fibre is a linear functional on the tangent space.[2] Symbolically,
where is linear.
Often one-forms are described locally, particularly in local coordinates. In a local coordinate system, a one-form is a linear combination of the differentials of the coordinates:
where the are smooth functions. From this perspective, a one-form has a covariant transformation law on passing from one coordinate system to another. Thus a one-form is an order 1 covariant tensor field.