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 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.



The most basic non-trivial differential one-form is the "change in angle" form   This is defined as the derivative of the angle "function"   (which is only defined up to an additive constant), which can be explicitly defined in terms of the atan2 function. Taking the derivative yields the following formula for the total derivative:   While the angle "function" cannot be continuously defined – the function atan2 is discontinuous along the negative  -axis – which reflects the fact that angle cannot be continuously defined, this derivative is continuously defined except at the origin, reflecting the fact that infinitesimal (and indeed local) changes in angle can be defined everywhere except the origin. Integrating this derivative along a path gives the total change in angle over the path, and integrating over a closed loop gives the winding number times  

In the language of differential geometry, this derivative is a one-form on the punctured plane. It is closed (its exterior derivative is zero) but not exact, meaning that it is not the derivative of a 0-form (that is, a function): the angle   is not a globally defined smooth function on the entire punctured plane. In fact, this form generates the first de Rham cohomology of the punctured plane. This is the most basic example of such a form, and it is fundamental in differential geometry.

Differential of a function


Let   be open (for example, an interval  ), and consider a differentiable function   with derivative   The differential   assigns to each point   a linear map from the tangent space   to the real numbers. In this case, each tangent space is naturally identifiable with the real number line, and the linear map   in question is given by scaling by   This is the simplest example of a differential (one-)form.

See also

  • Differential form – Expression that may be integrated over a region
  • Inner product – Generalization of the dot product; used to define Hilbert spaces
  • Reciprocal lattice – Fourier transform of a real-space lattice, important in solid-state physics
  • Tensor – Algebraic object with geometric applications


  1. ^ "2 Introducing Differential Geometry‣ General Relativity by David Tong". Retrieved 2022-10-04.
  2. ^ McInerney, Andrew (2013-07-09). First Steps in Differential Geometry: Riemannian, Contact, Symplectic. Springer Science & Business Media. pp. 136–155. ISBN 978-1-4614-7732-7.