One-form (differential geometry)

In differential geometry, a one-form (or covector field) on a differentiable manifold is a differential form of degree one, that is, a smooth section of the cotangent bundle.^[1] Equivalently, a one-form on a manifold ${\displaystyle M}$ is a smooth mapping of the total space of the tangent bundle of ${\displaystyle M}$ to ${\displaystyle \mathbb {R} }$ whose restriction to each fibre is a linear functional on the tangent space.^[2] Symbolically,

$${\displaystyle \alpha :TM\rightarrow {\mathbb {R} },\quad \alpha _{x}=\alpha |_{T_{x}M}:T_{x}M\rightarrow {\mathbb {R} },}$$ where ${\displaystyle \alpha _{x}}$ 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: $${\displaystyle \alpha _{x}=f_{1}(x)\,dx_{1}+f_{2}(x)\,dx_{2}+\cdots +f_{n}(x)\,dx_{n},}$$ where the ${\displaystyle f_{i}}$ 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.