One-form

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In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space. For details, see linear functional.

In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Explicitly, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to R whose restriction to each fibre is a linear functional on the tangent space. Symbolically,

\alpha : TM \rightarrow {\mathbf R},\quad \alpha_x = \alpha|_{T_xM}: T_xM\rightarrow {\mathbf R}

where αx 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:

\alpha_x = f_1(x)dx^1+f_2(x)dx^2+\dots+f_n(x)dx^n

where the fi are smooth functions. Note the use of upper numerical indices, not to be confused with powers. From this perspective, a one-form has a covariant transformation law on passing from one coordinate system to another. Thus a one-form is a rank 1 covariant tensor field.

Special cases

Let U \subseteq \mathbb{R} be open (e.g. an interval (a,b)), and consider a differentiable function f: U \to \mathbb{R} , with derivative f'. The differential df of f, at a point x_0\in U , is defined as a certain linear map of the variable dx. Specifically, df(x_0, dx): dx \mapsto f'(x_0) dx . (The meaning of the symbol dx is thus revealed: it is simply an argument, or independent variable, of the function df.) Hence the map x \mapsto df(x,dx) sends each point x to a linear functional df(x,dx). This is the simplest example of a differential (one-)form.

In terms of the de Rham complex, one has an assignment from zero-forms (scalar functions) to one-forms i.e. f\mapsto df.

A one-form is said to be a closed one-form if it is differentiable and its exterior derivative is everywhere equal to zero.

See also

References

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  • This page was last modified on 3 June 2008, at 15:15.

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