One-form

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. 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. Let ${\displaystyle U}$ be an open subset of ${\displaystyle M}$ and ${\displaystyle p\in U}$. Then $${\displaystyle {\begin{aligned}\omega :U&\rightarrow \bigcup _{p\in U}T_{p}^{*}(M)\\p&\mapsto \omega _{p}\in T_{p}^{*}(M)\end{aligned}}}$$ defines a one-form ${\displaystyle \omega }$. ${\displaystyle \omega _{p}}$ is a covector.

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.